{ "status": "complete-scan-compared-source-map", "pages": [ { "source": "book-05", "pdfPage": 1, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fa9d6905b5ee2fa6a026a5bd5b3313793137053ad8bb1ccef33e9f2f9d9e7e2f", "textHash": "e50a00ca6e9f4dfd7c42d241f0cf5e82af1aa953835987c3415fd48525313fbd", "characters": 74, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0001.png", "sha256": "f79fe9cdced2640267c91fd240cd80cb4f6ebf05dc9c9c37c92e6a6229e31165" } }, { "source": "book-05", "pdfPage": 2, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "68074786fbbadbb268de25ff1ddf611e7f09f7fac9d13251e1f08f1e5b627572", "textHash": "392bf28f906cb863f051f3f7fd497e7c2911ace96d1f0d1c0ffc50beb57099ff", "characters": 1622, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0002.png", "sha256": "ed7e2a5b0f8f8e218c9147a4bebdda106a65df076d1bbb07ce45cb6cae6f3b7b" } }, { "source": "book-05", "pdfPage": 3, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "219ce38f4f7aed640be9987d6ff5ec000f52e3e8192d6407b8321b9d670d9ecf", "textHash": "e312f2b685cdee15f91d87dfdad02a3a7a0e7820fe46e15e22d04e5828aa4b92", "characters": 81, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0003.png", "sha256": "9c1c8e9bc2963e9aff95fd09a21db0b35e4d8d0882fe409819be546db019b407" } }, { "source": "book-05", "pdfPage": 4, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "db14ec0ca6632b7f7ff94116a8d4dae92f6e49f285ef806c554cfea5fd9d0f52", "textHash": "a0f972ba8dae2d799a2bc767334565c6fa39d0411305b6291c08dd815fd84d3f", "characters": 1617, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0004.png", "sha256": "10dd6b667134daa5838f989345bd64cea948ede7244633dd7fcabe708c5bf463" } }, { "source": "book-05", "pdfPage": 5, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8c5d7757c785a5b7c816a3f80d0c4628958d34e1e14006f7c6d31fa9816f0a41", "textHash": "a2bc7a9f091bebf989fce6b664dd33901b15b7718260f0889b3d9a4e38c859e3", "characters": 2037, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0005.png", "sha256": "5766676c0d7fac52bac0cf309f78da39856f7c10357cc0aafcc387c4a2e37d5f" } }, { "source": "book-05", "pdfPage": 6, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8ac7f47cd2629e7c6b377a679af68aa96016784f93db481f3435c13152be5def", "textHash": "2f846adf8c3dd0b1fee6e2aa4d18b60eefc52485968c138622828afde4b4567e", "characters": 1625, "flags": [], "candidateHeadings": [ "Concept: This will be a general problem-solving technique or strategy. These are the \"keys\" to becoming a better problem solver!", "Important: This will be something important that you should learn. It might be a formula, a solution technique, or a caution." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0006.png", "sha256": "56edbc3622e24c688b8c7784f7601c1f8d69ee1bc3ccdab9d49345dbdf2e433f" } }, { "source": "book-05", "pdfPage": 7, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e74fe97018e7c82c104e7529ce6bee95cb0979ec1b44d2cad28109f36e150f51", "textHash": "41b2528161aa0cdab57bd2b977e0089f824c4bc7f4bd6460fd74f02be2afbaca", "characters": 2279, "flags": [], "candidateHeadings": [ "Exercises, Review Problems, and Challenge Problems" ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0007.png", "sha256": "42393d4dfce4bb96c7341cea395258486442217b1ebcd3444a469598880e301b" } }, { "source": "book-05", "pdfPage": 8, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d107cb94001346ceb653be4dc0fedd19c126fde95f7b1035b978cfd8f00d7512", "textHash": "be6c43a53bbe78645d58708e23d7a68893d37b6e13c4e1177eeba9189efd44b9", "characters": 2250, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0008.png", "sha256": "80c27e16c5ca835d5dba601a246c392beddda1e4f853873f9072094a7d1d0295" } }, { "source": "book-05", "pdfPage": 9, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c86e612d621494436e4ebf7d2e65875f02db3a0df848cefba0b9dec4065c9d92", "textHash": "60e4dace0ad8f7fb9219ca6b28656120eb68ebfb19ddfebf6968c85f0f8bb40b", "characters": 2878, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "educational-frontmatter", "evidence": "book05-source/page-0009.png", "sha256": "bcc43a5cc3f2d4a4b0e8d9d922df934d19ae2db5a6675c8c0793ff997c4579eb" } }, { "source": "book-05", "pdfPage": 10, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "cdde0ebcf25cc173b9a3935a9e113a5085289629513aed14db7f6c7f9175b46e", "textHash": "b0e75378dc5f1386c53253221fd827728c37f26f015335bae107c270ab1f5eba", "characters": 2453, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0010.png", "sha256": "cb1800619013950dc888c828b9f20683d13394868f56f4073023b42ab450c51f" } }, { "source": "book-05", "pdfPage": 11, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "728d68c50ad5c8dbc0be4a0383999bed0d9ef234f56d232df6b69b3a58bd98b4", "textHash": "427aa7758133fa68987ecc8b05c55b2efcf858cbbb26c89990b5bbfbc207e65f", "characters": 2713, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0011.png", "sha256": "27a34555907293a6520c5575dcfd47f4b741d4c0aed6a258ade130fb552b0419" } }, { "source": "book-05", "pdfPage": 12, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "497cba4d175ce02624edb9f4156ed207cb22c9beb298250bded778600aabf762", "textHash": "a5926764eda4f6e69344fd0622704ffa744c84973475bd6dca68770b1bb2df9c", "characters": 527, "flags": [], "candidateHeadings": [ "1.1 Introduction", "1.2 Making Integers Out of Integers", "1.3 Integer Multiples", "1.4 Divisibility of Integers", "1.5 Divisors", "1.6 Using Divisors", "1.7 Mathematical Symbols", "1.8 Summary", "2.1 Introduction", "2.2 Primes and Composites", "2.3 Identifying Primes I", "2.4 Identifying Primes II", "2.5 Summary", "3.1 Introduction", "3.2 Common Divisors" ], "candidateMarkers": [ { "id": "1.1", "label": null, "offset": 109 }, { "id": "1.2", "label": null, "offset": 128 }, { "id": "1.3", "label": null, "offset": 166 }, { "id": "1.4", "label": null, "offset": 190 }, { "id": "1.5", "label": null, "offset": 222 }, { "id": "1.6", "label": null, "offset": 238 }, { "id": "1.7", "label": null, "offset": 260 }, { "id": "1.8", "label": null, "offset": 288 }, { "id": "2.1", "label": null, "offset": 331 }, { "id": "2.2", "label": null, "offset": 351 }, { "id": "2.3", "label": null, "offset": 380 }, { "id": "2.4", "label": null, "offset": 408 }, { "id": "2.5", "label": null, "offset": 437 }, { "id": "3.1", "label": null, "offset": 481 }, { "id": "3.2", "label": null, "offset": 501 } ], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0012.png", "sha256": "2975e32f58b83bc1c9b2fb456303d8d37dade423eb359d9a27f79fb8f6fb4620" } }, { "source": "book-05", "pdfPage": 13, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d6543a7fba0944da7b01c2c9ac2066f67cae5f75cfb9f1c13e05e225a9926952", "textHash": "c770eee259333bd19c3736b50995cabc18d9b6028621be50c9a72056f0577e95", "characters": 1043, "flags": [], "candidateHeadings": [ "3.3 Greatest Common Divisors (GCDs) ..... 41", "3.4 Common Multiples ..... 43", "3.5 Remainders ..... 46", "3.6 Multiples, Divisors, and Arithmetic ..... 50", "3.7 The Euclidean Algorithm ..... 53", "3.8 Summary ..... 60", "4.1 Introduction ..... 63", "4.2 Factor Trees ..... 64", "4.3 Factorization and Multiples ..... 68", "4.4 Factorization and Divisors ..... 72", "4.5 Rational Numbers and Lowest Terms ..... 75", "4.6 Prime Factorization and Problem Solving ..... 78", "4.7 Relationships Between LCMs and GCDs ..... 80", "4.8 Summary ..... 84", "5.1 Introduction ..... 91", "5.2 Counting Divisors ..... 91", "5.5 Summary ..... 104", "6.1 Introduction ..... 109", "6.2 Some Special Primes ..... 109", "6.3 Factorials, Exponents and Divisibility ..... 111", "6.4 Perfect, Abundant, and Deficient Numbers ..... 115", "6.5 Palindromes ..... 117", "6.6 Summary ..... 120", "7.1 Introduction ..... 125" ], "candidateMarkers": [ { "id": "3.3", "label": null, "offset": 9 }, { "id": "3.4", "label": null, "offset": 55 }, { "id": "3.5", "label": null, "offset": 85 }, { "id": "3.6", "label": null, "offset": 109 }, { "id": "3.7", "label": null, "offset": 158 }, { "id": "3.8", "label": null, "offset": 195 }, { "id": "4.1", "label": null, "offset": 248 }, { "id": "4.2", "label": null, "offset": 274 }, { "id": "4.3", "label": null, "offset": 300 }, { "id": "4.4", "label": null, "offset": 341 }, { "id": "4.5", "label": null, "offset": 381 }, { "id": "4.6", "label": null, "offset": 428 }, { "id": "4.7", "label": null, "offset": 481 }, { "id": "4.8", "label": null, "offset": 530 }, { "id": "5.1", "label": null, "offset": 580 }, { "id": "5.2", "label": null, "offset": 606 }, { "id": "5.3", "label": null, "offset": 637 }, { "id": "5.4", "label": null, "offset": 677 }, { "id": "5.5", "label": null, "offset": 709 }, { "id": "6.1", "label": null, "offset": 760 }, { "id": "6.2", "label": null, "offset": 787 }, { "id": "6.3", "label": null, "offset": 821 }, { "id": "6.4", "label": null, "offset": 874 }, { "id": "6.5", "label": null, "offset": 929 }, { "id": "6.6", "label": null, "offset": 955 }, { "id": "7.1", "label": null, "offset": 1012 } ], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0013.png", "sha256": "7eedfcee603acb5bb04747436bade4421d1e6d1c61fde7d98740628f3b2fc223" } }, { "source": "book-05", "pdfPage": 14, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f99c38e99f89aed770fe33eb98c04d2d4cf7d09c6fac4e34777da330ab41cada", "textHash": "79b89307eeb06392ffbff04d4c739923b9e8b525764debaf0428a4f539673de3", "characters": 1030, "flags": [], "candidateHeadings": [ "7.2 Problems ..... 125", "7.3 Summary ..... 137", "8.1 Introduction ..... 141", "8.2 Counting in Bundles ..... 141", "8.3 Base Numbers ..... 145", "8.4 Base Number Digits ..... 148", "8.5 Converting Integers Between Bases ..... 150", "8.7 Summary ..... 161", "9.1 Introduction ..... 165", "9.2 Base Number Addition ..... 165", "9.3 Base Number Subtraction ..... 168", "9.4 Base Number Multiplication ..... 170", "9.5 Base Number Division and Divisibility ..... 172", "9.6 Summary ..... 175", "10.1 Introduction ..... 177", "10.2 Units Digits in Arithmetic ..... 177", "10.3 Base Number Units Digits ..... 184", "10.4 Unit Digits Everywhere! ..... 187", "10.5 Summary ..... 190", "11.1 Introduction ..... 195", "11.2 Terminating Decimals ..... 195", "11.3 Repeating Decimals ..... 201", "11.4 Converting Decimals to Fractions ..... 205", "11.6 Summary ..... 212" ], "candidateMarkers": [ { "id": "7.2", "label": null, "offset": 9 }, { "id": "7.3", "label": null, "offset": 33 }, { "id": "8.1", "label": null, "offset": 81 }, { "id": "8.2", "label": null, "offset": 108 }, { "id": "8.3", "label": null, "offset": 142 }, { "id": "8.4", "label": null, "offset": 169 }, { "id": "8.5", "label": null, "offset": 202 }, { "id": "8.6", "label": null, "offset": 250 }, { "id": "8.7", "label": null, "offset": 294 }, { "id": "9.1", "label": null, "offset": 352 }, { "id": "9.2", "label": null, "offset": 379 }, { "id": "9.3", "label": null, "offset": 414 }, { "id": "9.4", "label": null, "offset": 452 }, { "id": "9.5", "label": null, "offset": 493 }, { "id": "9.6", "label": null, "offset": 545 }, { "id": "10.1", "label": null, "offset": 594 }, { "id": "10.2", "label": null, "offset": 622 }, { "id": "10.3", "label": null, "offset": 664 }, { "id": "10.4", "label": null, "offset": 704 }, { "id": "10.5", "label": null, "offset": 743 }, { "id": "11.1", "label": null, "offset": 803 }, { "id": "11.2", "label": null, "offset": 831 }, { "id": "11.3", "label": null, "offset": 867 }, { "id": "11.4", "label": null, "offset": 901 }, { "id": "11.5", "label": null, "offset": 949 }, { "id": "11.6", "label": null, "offset": 1002 } ], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0014.png", "sha256": "06cc2203fb5dd98ea5015056d3e960dfc8f400d382eee222349441f9a2b6d4b5" } }, { "source": "book-05", "pdfPage": 15, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "03235f5a83fffab8c93b9f90a1e59e6a8261ae7bfe07332cf39a236d5312db6d", "textHash": "d55ee776c638423c1569b34dafcb85e4323e2b6eb4107e9879a607d25619d090", "characters": 863, "flags": [], "candidateHeadings": [ "12.1 Introduction", "12.2 Congruence", "12.3 Residues", "12.4 Addition and Subtraction", "12.5 Multiplication and Exponentiation", "12.6 Patterns and Exploration", "12.7 Summary", "13.1 Introduction", "13.2 Divisibility Rules", "13.4 Summary", "14.1 Introduction", "14.2 Modular Inverses and Simple Linear Congruences", "14.3 Solving Linear Congruences", "14.4 Systems of Linear Congruences", "14.5 Summary", "15.1 Introduction", "15.2 Familiar Factors and Divisibility", "15.3 Algebraic Methods of Arithmetic", "15.4 Useful Forms of Numbers", "15.5 Simplicity", "15.6 Summary" ], "candidateMarkers": [ { "id": "12.1", "label": null, "offset": 48 }, { "id": "12.2", "label": null, "offset": 66 }, { "id": "12.3", "label": null, "offset": 82 }, { "id": "12.4", "label": null, "offset": 96 }, { "id": "12.5", "label": null, "offset": 126 }, { "id": "12.6", "label": null, "offset": 165 }, { "id": "12.7", "label": null, "offset": 195 }, { "id": "13.1", "label": null, "offset": 231 }, { "id": "13.2", "label": null, "offset": 249 }, { "id": "13.3", "label": null, "offset": 273 }, { "id": "13.4", "label": null, "offset": 311 }, { "id": "14.1", "label": null, "offset": 347 }, { "id": "14.2", "label": null, "offset": 365 }, { "id": "14.3", "label": null, "offset": 417 }, { "id": "14.4", "label": null, "offset": 449 }, { "id": "14.5", "label": null, "offset": 484 }, { "id": "15.1", "label": null, "offset": 514 }, { "id": "15.2", "label": null, "offset": 532 }, { "id": "15.3", "label": null, "offset": 571 }, { "id": "15.4", "label": null, "offset": 608 }, { "id": "15.5", "label": null, "offset": 637 }, { "id": "15.6", "label": null, "offset": 653 } ], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0015.png", "sha256": "b5f65d4dfbde4f9fb3c2160a853038cb4bf21af2b5673029d20f6b90b6f2bf46" } }, { "source": "book-05", "pdfPage": 16, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "72e4d2a354977a7c76f42793b27d15832896b7de06451221459391b94d0a673e", "textHash": "a4f0d36826510ca98c2a4c5b8e459b449231dc7bab5b596a02a4defa73542a06", "characters": 991, "flags": [], "candidateHeadings": [ "1.1 Introduction", "Definitions: The integers can be divided into three groups: positive integers, negative integers and zero.", "Important: A number with at least one non-zero digit after its decimal point is not an integer because we cannot count from zero to that number (forward or backward) 1 unit at a time on the number line." ], "candidateMarkers": [ { "id": "1.1", "label": null, "offset": 119 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0016.png", "sha256": "874a67f25ae989cd8398a3aec2bf6c6804b19deee74f9fa977b72286f4b0b619", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 17, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "041ff3fc28bdcfb5ca12961e7577295153936ac26b2b958ffd3b16afce176def", "textHash": "7ff4e04511771837aaf46b9c79003b6c17102f46f57e5a3517da962ea979be23", "characters": 1555, "flags": [], "candidateHeadings": [ "Concept: Integers are the numbers that we use to count both forward and backward when counting whole objects such as people, pennies, cars, etc.", "Definitions: Positive integers are often called natural numbers and you should become familiar with that term if you are not already. We also sometimes call positive integers counting numbers because we use them to count objects. Sometimes we refer to whole numbers, which include 0 and all positive integers. Most of the time this book deals with natural numbers, though much of the material also applies to 0 and negative numbers.", "Concept: Negative integers describe the backward counting that results when subtracting objects from a group." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0017.png", "sha256": "113369ba1d82e502c7b9271c1e045cc73ce937c589d9f4e8fdf7143b4b73cf36", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 18, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "17e20f1271a169de0c1d485403a231a507c19262436ceca3ed09099e3c1b7d91", "textHash": "e8a5c7d6a9ef25f5bd28dc165f8c8e4ae4605e98230d4113ebffa8e172ac2b43", "characters": 2268, "flags": [], "candidateHeadings": [ "1.2 Making Integers Out of Integers" ], "candidateMarkers": [ { "id": "1.2", "label": null, "offset": 37 }, { "id": "1.1", "label": "Problem", "offset": 376 }, { "id": "2.4", "label": null, "offset": 560 }, { "id": "1.2", "label": "Problem", "offset": 588 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0018.png", "sha256": "6e2f7dad41d16291e04d9baa88e4ab0d3f5511e244e2057b18788c95ec39c7da", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 19, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6c666317334100dbb5638ecd38929944bfafd10a1bf8159e8cea7da2591b18a7", "textHash": "9d5fc50989ea425a4c1ee52c72fe766e63e221448188f4b2b490d3d26706aca2", "characters": 8297, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "1.1", "label": "Problem", "offset": 32 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0019.png", "sha256": "8bfe6f92ed4d92d230407f1e02dfb09159784e67a6024ad87754ad4efab0ff05", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 20, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0170fa374df4188541fc00d34ce3e35b7a05713287925352b04e67d6ccb55588", "textHash": "4a278cf67d1aeb246cb013801596f1ba44ce61e176ea1dbda1d56a99bf01ac17", "characters": 6068, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0020.png", "sha256": "3300c96bfeb49d9a13d66152e601d355559936816db0c03fb44e7105a4eda409", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 21, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3ec669bdef6aecb2e43c78fed1bb5df4d2b6a3e83fab9a2353ff4dbc2b2102c8", "textHash": "ef8913135cc738ce2457726d2b5257f9c75866cc439e7e69e53de5569323c99f", "characters": 1513, "flags": [], "candidateHeadings": [ "Definitions:", "Exercises" ], "candidateMarkers": [ { "id": "1.2", "label": "Problem", "offset": 32 }, { "id": "1.2", "label": "Solution for Problem", "offset": 143 }, { "id": "1.2.1", "label": null, "offset": 1207 }, { "id": "1.2.2", "label": null, "offset": 1300 }, { "id": "1.2.3", "label": null, "offset": 1369 }, { "id": "1.2.4", "label": null, "offset": 1460 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0021.png", "sha256": "2758ed0313da67b8fc003f8cb2f3d59a7e508ba6a51a943affdd841811971763", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 22, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b9a04167f4cb227eb6b14c202d87f9075b55d64aeece9c6fc794b75fb5fe1f30", "textHash": "630fba702d14a80ea4793b9fec23184f16cce7f32e6f9c29fc2b6202eb1e5729", "characters": 1326, "flags": [], "candidateHeadings": [ "1.3 Integer Multiples" ], "candidateMarkers": [ { "id": "1.3", "label": null, "offset": 23 }, { "id": "1.3", "label": "Problem", "offset": 347 }, { "id": "1.4", "label": "Problem", "offset": 448 }, { "id": "1.5", "label": "Problem", "offset": 752 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0022.png", "sha256": "bc52940014a4cdb0634696051ab47fe056a296d9455bf08566b9f6176dd1037d", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 23, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5d3eb0622d880e131f7907fd88c8a08de4ce4f3cfbed0b19af627b66f3c887c4", "textHash": "7b73aaa2a625880466e0eeb25b6c1afac5a90ade9bc978636edbfc2034dbbad8", "characters": 6372, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "1.3", "label": "Problem", "offset": 32 }, { "id": "1.3", "label": "Solution for Problem", "offset": 132 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0023.png", "sha256": "a274a9a9538903f4e86b6c7083f3f287e6a3692bc52978acdefc7ced98b96fac", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 24, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b278bcbfe6eb9bf2a6adaa41c1523e13208143731277ce66ae01f5f61bf6b709", "textHash": "998613e4841ba28e5ee3a870bc715dd83d97b01bcd9c748345961c4535123d65", "characters": 1982, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "1.0", "label": null, "offset": 894 }, { "id": "1.0", "label": null, "offset": 898 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0024.png", "sha256": "f3cdeb428db3eae9ed1aa2113e62e99d095d59b3469385036406d9460272efc6", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 25, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e6dcd21b1b5b0c53afed2fc719163cbd177176fad72a34c18df50ebf51e202bb", "textHash": "30bb9dce495c427a82605407ccbaf566e0a7915331bd94ae8ec4aafb76718180", "characters": 1500, "flags": [], "candidateHeadings": [ "3.2 = 6 3.3 = 9 3.4 = 12", "Definition: A multiple of an integer is the product of that integer with any integer.", "Important: For any integers m and n, their product mn is both a multiple of m and a multiple of n.", "Exercises" ], "candidateMarkers": [ { "id": "3.2", "label": null, "offset": 160 }, { "id": "1.5", "label": "Problem", "offset": 774 }, { "id": "1.5", "label": "Solution for Problem", "offset": 944 }, { "id": "1.3.1", "label": null, "offset": 1379 }, { "id": "1.3.2", "label": null, "offset": 1408 }, { "id": "1.3.3", "label": null, "offset": 1437 }, { "id": "1.3.4", "label": null, "offset": 1467 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0025.png", "sha256": "d3bf7e03d45aad1a6241c255550749f70aebfa22e77cc57065bd20b898635f8b", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 26, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "93df0f7975d5d890cbb7289992d2fd6c70cf086dcd03484e2559c03638253a13", "textHash": "4091ebdd4853917994a8a6b939b4fc22f78c54519704a793f60dd0c95d7aa8b3", "characters": 1619, "flags": [], "candidateHeadings": [ "1.4 Divisibility of Integers" ], "candidateMarkers": [ { "id": "1.3.5", "label": null, "offset": 30 }, { "id": "1.3.6", "label": null, "offset": 83 }, { "id": "1.3.7", "label": null, "offset": 137 }, { "id": "1.3.8", "label": null, "offset": 190 }, { "id": "1.3.9", "label": null, "offset": 242 }, { "id": "1.3.10", "label": null, "offset": 424 }, { "id": "1.4", "label": null, "offset": 503 }, { "id": "1.6", "label": "Problem", "offset": 729 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0026.png", "sha256": "c863cafa5cd9467c6c22ed5d559e960e62c66a13eae62a7113c5b2478507110e", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 27, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "56b2b8d927e6f43753a05f9ac097504ff08f0ff5e19fcaba1135288a7152e261", "textHash": "c7e593eabca94da734a506b5adf68ed4e2b064adf24be1d13f5a7f1a694d4826", "characters": 9768, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "1.6", "label": "Problem", "offset": 32 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0027.png", "sha256": "76c763b8c016de4a2cdaeb429b61ae07bf2ad73b35247962aedbfec695cf800e", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 28, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f40fcfb7e3cbe18000954ab77d3ed6b9ec6f387f13343e5e3c84d834b07cb54a", "textHash": "ea87e412da72019e6720fbb9cb99720236909948155d1455f5650ead1d7fc3d5", "characters": 1276, "flags": [], "candidateHeadings": [ "Important: When \\(n \\neq 0\\), the statement, \"m is divisible by n,\" means exactly the same thing as the statement, \"m is a multiple of n.\"" ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0028.png", "sha256": "b4470b246502844028baf34b5267461a8a3a5c424e35248cfa3cbb9862096b69", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 29, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e6f64b3f97b2c4b7005e4f918af54dc062ef32767de7af7e5b7e4943cb0ee75e", "textHash": "af4bcfe4a766678b76e59ad20ad587e9efb7ed83d68f4e450d10170fd8e35613", "characters": 1081, "flags": [], "candidateHeadings": [ "1.5 Divisors" ], "candidateMarkers": [ { "id": "1.4.1", "label": null, "offset": 32 }, { "id": "1.4.2", "label": null, "offset": 158 }, { "id": "1.4.3", "label": null, "offset": 285 }, { "id": "1.4.4", "label": null, "offset": 313 }, { "id": "1.4.5", "label": null, "offset": 340 }, { "id": "1.4.6", "label": null, "offset": 368 }, { "id": "1.4.7", "label": null, "offset": 396 }, { "id": "1.4.8", "label": null, "offset": 425 }, { "id": "1.4.9", "label": null, "offset": 455 }, { "id": "1.5", "label": null, "offset": 485 }, { "id": "1.7", "label": "Problem", "offset": 694 }, { "id": "1.8", "label": "Problem", "offset": 836 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0029.png", "sha256": "76768bcca037a5209ecd4546f92a5356436995d6830cc04625aa2ac6c150d4bb", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 30, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "49727929fde9d2f71f0eb1958edc8fd9771d3cd0547cda2c75473bc66417366c", "textHash": "823a80c80dec780124c58f6485f1952fa802c0e780eca5c61fd52a9834f9c82b", "characters": 1692, "flags": [], "candidateHeadings": [ "Definition: When an integer n is divisible by another integer d, we say that d is a divisor of n.", "Important: Negative integers can be divisors too! One of the divisors of 6 is -2 because \\(\\frac{6}{-2} = -3\\),", "Important: When \\(d \\neq 0\\), the following statements all mean the same thing:", "Important: For an integer n we know that" ], "candidateMarkers": [ { "id": "1.7", "label": "Problem", "offset": 466 }, { "id": "1.7", "label": "Solution for Problem", "offset": 609 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0030.png", "sha256": "31f0f4cad9dd14185f1c96820b2a9b34b0ffce5c737da1a2c91b20b9ce4a1510", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 31, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "312804207206396a5f78b01028332e6909a1873479fca172685b3d71f65315ed", "textHash": "816ee75c0488754028f7a710aa917f0c2dbecfdceeae48c8cdd7b0f8d9c57f25", "characters": 1983, "flags": [], "candidateHeadings": [ "Definition: A proper divisor of an integer n is a positive divisor of n other than n itself.", "Important: An interesting and sometimes useful fact is that every integer other than 0 is a divisor of 0. We know this is true because for any integer n ≠ 0,", "Important: Since there are no multiples of a positive integer n between 0 and n, we know that n cannot be a divisor of any integer between 0 and n. This means that when hunting for the positive proper divisors of a positive integer m, we must only test the integers between 0 and m. For instance, we need not test to see whether 36 is a divisor of 14 because" ], "candidateMarkers": [ { "id": "1.8", "label": "Problem", "offset": 1225 }, { "id": "1.8", "label": "Solution for Problem", "offset": 1437 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0031.png", "sha256": "0b806264c3e0a41b282f583f704bae6162e115fc99922f21800f2f44dbf9c702", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 32, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e8d881bf7f024f1ac9527183799e7a71e9fd3d81d9efcf8e062404b1d1d46f3d", "textHash": "c5eb7f370c62d3e489c0304e3603dce36ce224c8045141e882044a591644b59c", "characters": 2166, "flags": [], "candidateHeadings": [ "Important: The following statements all mean the same thing. More importantly, understanding them will help us solve many number theory problems.", "Concept: It is often helpful in mathematics to see the same facts expressed in multiple ways. This helps us understand ideas better and apply them to a wider variety of problems." ], "candidateMarkers": [ { "id": "1.8", "label": "Problem", "offset": 1270 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0032.png", "sha256": "82f2cdd7171a76d9f545db2967c7afc1fba9c39ef2965eb76a586ad41230a01d", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 33, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f73e683861a5746a63efceec0736d1a5e9ff15c66b6f69132c799a95ea52abc2", "textHash": "537799537f22ce4b3667fb131fb456234caddf722f8908e8053ea03925cf8409", "characters": 2063, "flags": [], "candidateHeadings": [ "Exercises", "1.6 Using Divisors" ], "candidateMarkers": [ { "id": "1.5.1", "label": null, "offset": 43 }, { "id": "1.6", "label": null, "offset": 200 }, { "id": "1.9", "label": "Problem", "offset": 364 }, { "id": "1.10", "label": "Problem", "offset": 699 }, { "id": "1.9", "label": "Problem", "offset": 1269 }, { "id": "1.9", "label": "Solution for Problem", "offset": 1429 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0033.png", "sha256": "601d1136966db9180e68294c7c5a80f0040ea08a16df62f081c8a52051bd1fac", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 34, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2022e050d8d2e1dc4f4852d2272bff4438a054cfa486fc47922d7a6eb84982de", "textHash": "559e235da0f4e80a5217472469d427361bd7586e5fc69ed07420d6bbde079f99", "characters": 2468, "flags": [], "candidateHeadings": [ "Concept: Begin working word problems by translating them into the language of math. This allows you to apply mathematical tools to the problem.", "Exercises" ], "candidateMarkers": [ { "id": "1.10", "label": "Problem", "offset": 711 }, { "id": "1.10", "label": "Solution for Problem", "offset": 1098 }, { "id": "1.6.1", "label": null, "offset": 1881 }, { "id": "1.6.2", "label": null, "offset": 2075 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0034.png", "sha256": "d348b3a42f8e5730dc9445cae29a37273d2d687c681b769309483f54816d92f1", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 35, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9e7197528d3c467a1cd07ddc2903c2f87a4fc73b044f19c54bea89f40df32428", "textHash": "5c66113843cfb68ae139665b4d6ebbec3ae26f2817600e644a0d8bab4af36ede", "characters": 1802, "flags": [], "candidateHeadings": [ "1.7 Mathematical Symbols", "Concept: One of the goals of mathematics is to simplify difficult ideas so that we can understand and use them more easily.", "Exercises" ], "candidateMarkers": [ { "id": "1.7", "label": null, "offset": 32 }, { "id": "1.7.1", "label": null, "offset": 1442 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0035.png", "sha256": "1b99a3e7c5084b3decea1badc671a30be1f4a60d244c6e36e8edcd47b76239ea", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 36, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "34d7850cc3df30a69cea2124f686c22f3bcfdab76d48b4f7a192e8373080f7a4", "textHash": "29ed0b3b34a8a4bf71d9767167c2599ce6f411ade93a4cedb43b629fa1e85cf4", "characters": 1799, "flags": [], "candidateHeadings": [ "1.8 Summary", "Definitions:", "Important: For any integers m and n, their product mn is both a multiple of m and a multiple of n.", "Important: When \\(d \\neq 0\\), the following statements all mean the same thing:" ], "candidateMarkers": [ { "id": "1.8", "label": null, "offset": 0 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0036.png", "sha256": "5afc061e7778ccd38443620b68e88b09a9f819e84d5f89ce9553ea687887920a", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 37, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "bc02165641b99faaaf49dbb49b2ba59fb7b328811667896c2e92122ee101bc14", "textHash": "f6c490c0f1db262acefaf8529f32fa7dde8965a903d0b63a356a2465fca90e5d", "characters": 1954, "flags": [], "candidateHeadings": [ "Concept: Integers are the numbers we use for counting indivisible objects such as people, pennies, and cars. We can think of negative integers as the ones we use to count backward when subtracting objects from a group. A number with at least one non-zero digit after its decimal point is not an integer because we cannot count to that number (forward or backward) 1 unit at a time on the number line.", "Concept: Addition and subtraction are fast counting. Multiplication is fast addition. Exponentiation is fast multiplication. This means that addition, subtraction, and multiplication of integers are just methods of fast counting and that taking an integer to a power that is a whole number (except \\(0^0\\), which is undefined) is also fast counting.", "Concept: The following statements are all true and mean the same thing:", "Concept: It is often helpful in mathematics to see the same facts expressed in two different ways. This can both help us understand ideas better and apply them to a greater number of problems.", "Concept: Begin working word problems by translating them into the language of math. This allows you to apply the tools of math to solve the problem." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0037.png", "sha256": "9bb823de30d98a25a414da066561d76b76e87a6e0df6e756a4d8a140ed43d3c1", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 38, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9b14a0ebf6cc4b1fa7d9b430dd4ab2c311a982d99e53612d977212be33809921", "textHash": "e3192c281cbc0630368230b007701a7ed6a0b7109568986db5a1fbf18d164467", "characters": 1500, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Concept: One of the goals of mathematics is to simplify difficult ideas so that we can understand and use them more easily.", "REVIEW PROBLEMS", "1.11 Find all the perfect squares between 200 and 300.", "1.12 Answer each of the following.", "1.13 List the 8 smallest positive multiples of 9.", "1.14 Find all of the positive divisors of each of the following integers.", "1.15 State whether each of the following are true or false." ], "candidateMarkers": [ { "id": "1.11", "label": null, "offset": 839 }, { "id": "1.12", "label": null, "offset": 895 }, { "id": "1.13", "label": null, "offset": 1208 }, { "id": "1.14", "label": null, "offset": 1259 }, { "id": "1.15", "label": null, "offset": 1361 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0038.png", "sha256": "ba422b5cb301ac75e2106d8856e816b93cece12d1b7fd110233e7a6e0065cff7", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 39, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ede85c1998150bbc2d5cd07b184e9100991565a50e577fea9ef6d3c233b8bd46", "textHash": "7cb7b66e611f59fa1bfc913f11a7c508a85739e7dd95921d5e07c5da31be8fd9", "characters": 1266, "flags": [], "candidateHeadings": [ "Challenge Problems", "1.16 Lucius, Sarah, Brandon, and Scott take Algebra together. On their first exam, Lucius got a 94, Sarah got a 91, Brandon got a 95, and Scott's score was between 81 and 87 inclusive. If the average of all four of their scores is an integer, what was Scott's score on the first exam?", "1.17 Find all the perfect cubes between 1000 and 2000. Hints: 146", "1.18 What is the largest integer whose cube is less than 10,000? (Source: MATHCOUNTS) Hints: 41", "1.19 How many integers between 1 and 100 are perfect powers?", "1.20 How many integers 1-9 are divisors of the five-digit number 24,516? (Source: MATHCOUNTS)", "1.21 Is 11111 divisible by 41?", "1.22 Is 111111 divisible by 37?", "1.23 Find the smallest natural number that is not a divisor of 5040.", "1.24 The diameter of a circle is a whole number. The area of the circle is between 100 and 120 square units. What is the number of units in the circle's diameter? (Source: MATHCOUNTS) Hints: 116" ], "candidateMarkers": [ { "id": "1.16", "label": null, "offset": 52 }, { "id": "1.17", "label": null, "offset": 339 }, { "id": "1.18", "label": null, "offset": 407 }, { "id": "1.19", "label": null, "offset": 505 }, { "id": "1.20", "label": null, "offset": 568 }, { "id": "1.21", "label": null, "offset": 664 }, { "id": "1.22", "label": null, "offset": 697 }, { "id": "1.23", "label": null, "offset": 731 }, { "id": "1.24", "label": null, "offset": 802 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\page-0039.png", "sha256": "fda44c68067b975551c2d5f919dc825492376a763a18449ff78973b1be023846", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "book-05", "pdfPage": 40, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "df637ba5dcc22961a876396712e1a002ecb89c1d936f7ecf03ffa47b98c73dac", "textHash": "36d892ee6cd05a762aa7db0ba5777f54f245f8dcc7117c78b4ff6c48cdc40bb3", "characters": 1142, "flags": [], "candidateHeadings": [ "2.1 Introduction", "2.2 Primes and Composites" ], "candidateMarkers": [ { "id": "2.1", "label": null, "offset": 251 }, { "id": "2.2", "label": null, "offset": 822 }, { "id": "2.1", "label": "Problem", "offset": 859 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0040.png", "sha256": "a5fc8ac6f33ee77f9169ee15a4c9e8adc49a06576d508ed253db41308466e3ff", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 41, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4ae21cc7585908259d36f4c5d0a0470f9543bd5c89a4edb6440f0a8b36b6521a", "textHash": "4bba7dbe181b20a5cf52b702d7714b38f2638dc587f27346ae0a6ef3a9715bd4", "characters": 2207, "flags": [], "candidateHeadings": [ "Definition: A prime number (or simply prime) is a natural number \\(p > 1\\) whose only positive divisors are 1 and \\(p\\).", "Definition: A composite number (or simply composite) is a natural number \\(c\\) with some positive divisor besides 1 and \\(c\\) itself." ], "candidateMarkers": [ { "id": "2.1", "label": "Problem", "offset": 244 }, { "id": "2.1", "label": "Solution for Problem", "offset": 314 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0041.png", "sha256": "23303f0514ff3e32015e3f557bda9a675870f693869e81e961add51747444f99", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 42, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8e94eb4b177f6af2b9ea0fe2615ec16b1e6d51546b6b5637024d10b3c0d7af70", "textHash": "50d53ed75beed5da19a4336c1c77b491b28abba28c83ee0202f865e15d8efe97", "characters": 1458, "flags": [], "candidateHeadings": [ "Important: A composite number \\(c\\) can be written as the product of two natural numbers that are both between 1 and itself:", "Important: 1 is the only natural number that is neither prime nor composite. It has only one positive divisor—itself. This is one reason why 1 is a very special number.", "Exercises" ], "candidateMarkers": [ { "id": "2.2.1", "label": null, "offset": 685 }, { "id": "2.2.2", "label": null, "offset": 847 }, { "id": "2.2.3", "label": null, "offset": 895 }, { "id": "2.2.4", "label": null, "offset": 1278 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0042.png", "sha256": "f59296dc6e7b9688cd9f986fb1d6050080e3adc558dfca0679049ad6249f7458", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 43, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "cf85ccfb74888cf972edafc8eb0822ae6390fc559ee1462ffcd303af19a0b193", "textHash": "1de8981eb706d2772e34a523a71be403464a1398db525c95d337be3b1a3bd47a", "characters": 1274, "flags": [], "candidateHeadings": [ "2.3 Identifying Primes I" ], "candidateMarkers": [ { "id": "2.3", "label": null, "offset": 33 }, { "id": "2.2", "label": "Problem", "offset": 68 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0043.png", "sha256": "23a7e9868002ad159a5f77bdc060f845e439e2fd69bd74e0e1ca68d92dc8f5c6", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 44, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0f83b69baeed19c4aaa6fa2da417c11a27be9cd5b8756e7b9fc3e14f6dc88b0f", "textHash": "d4745b2dd0346127b5852ed572eff44788e3f7d638ae65060a1ccb0f240303a8", "characters": 1253, "flags": [], "candidateHeadings": [ "2.2 = 4", "3.2 = 6", "4.2 = 8", "5.2 = 10" ], "candidateMarkers": [ { "id": "2.2", "label": "Problem", "offset": 26 }, { "id": "2.2", "label": "Solution for Problem", "offset": 133 }, { "id": "2.2", "label": null, "offset": 925 }, { "id": "3.2", "label": null, "offset": 934 }, { "id": "4.2", "label": null, "offset": 942 }, { "id": "5.2", "label": null, "offset": 950 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0044.png", "sha256": "d3019ecf3e719fd998440e399fe76c04391e0b9436ffb270c9dcfbe052300099", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 45, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ec9fbfebef5240b6b8af1528a4f5a426bb76aaf2a6961e46b9481d16c54f03ae", "textHash": "5be4456ab37ee29a868fe339c23b02d33f1dc5cfc32a20bf6f9affbfec2fe469", "characters": 621, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0045.png", "sha256": "09d9fb05cbb12408b582610161468d7cd0047d4af7168c86cfd099ea41f3a5c0", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 46, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "171e3fd4b4d3b3d91af436bafc26e47d63d783d126a80cfa369093137d3e8d92", "textHash": "d5c78bf8ba8897ed9f0ab247586f7facd8354e95cd0e06873bbf2d4f9fbef186", "characters": 1853, "flags": [], "candidateHeadings": [ "Concept: Keep in mind that it is far more important to understand how the process of the Sieve works than to remember (or even spell!) Eratosthenes. Identifying and describing prime numbers are very important in mathematics. We will learn how to use primes to help solve all kinds of interesting problems whether or not we remember the names of ancient Greek mathematicians.", "Exercises", "2.4 Identifying Primes II" ], "candidateMarkers": [ { "id": "2.3.1", "label": null, "offset": 1388 }, { "id": "2.3.2", "label": null, "offset": 1508 }, { "id": "2.4", "label": null, "offset": 1599 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0046.png", "sha256": "5249c13af03ce2113d36a5e37080073ff2d60a40bd682d97f4a741c04b14719f", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 47, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b6af52101b168cd46f4afc297732baa911fcb196177c65561103e8036ece92d7", "textHash": "21c019b0642e595ba76f1594ee9d7d7f7be4ccc6373da9816ef501a2461ff594", "characters": 2025, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "2.3", "label": "Problem", "offset": 43 }, { "id": "2.4", "label": "Problem", "offset": 357 }, { "id": "2.5", "label": "Problem", "offset": 674 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0047.png", "sha256": "24e92a3f6077772758a6a9854a65ebd6f40d6897c984b479edb1030fbcd7255d", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 48, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "36dd08026110b68ae89ed007465c35e60c803606cd0ba76e42bfca10cc2529e7", "textHash": "56874a6f502a83f56e9caf2fc2f693c704c3ec7eb22421d6a9dc5485a10d63af", "characters": 2467, "flags": [], "candidateHeadings": [ "Important: When performing the Sieve of Eratosthenes, we cross out each composite number while crossing out all the multiples of its smallest prime divisor." ], "candidateMarkers": [ { "id": "2.3", "label": "Problem", "offset": 27 }, { "id": "2.3", "label": "Solution for Problem", "offset": 341 }, { "id": "2.4", "label": "Problem", "offset": 1450 }, { "id": "2.4", "label": "Solution for Problem", "offset": 1561 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0048.png", "sha256": "027fc61efd44189687baea554363ab9ad28816a290d6f7106c31ba5a8244e942", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 49, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "40e76bd2479e2e3bc24a810c9237f2eec98add87c32b947a8bbf6c17e3c5db5f", "textHash": "37f03931bff12b8af1139d078f8e8bb2e76faf0d143edb947efeb845fb04f88a", "characters": 3013, "flags": [], "candidateHeadings": [ "Concept: When faced with a challenging problem, play around with simple examples. They often provide clues that help us discover the solution to the whole problem.", "Concept: When faced with a difficult problem, finding a pattern often helps us guess the solution. A good guess helps us know what to look for when searching for a complete solution." ], "candidateMarkers": [ { "id": "2.4", "label": "Problem", "offset": 1675 }, { "id": "2.5", "label": "Problem", "offset": 2065 }, { "id": "2.5", "label": "Solution for Problem", "offset": 2258 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0049.png", "sha256": "6fd0bc5d11efc9a730cb434dc5b9c083d9ba8ad98b8271e584da3cb9ab86a0c1", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 50, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9ce3c6e9171fe18ec778fcef52f83446f744be63dba7ae4564ebbce34048b4f9", "textHash": "5b2dfc42ad8de50934e39e677cd59450c33a61ea22f34fec22c45d8550b1d0cc", "characters": 2491, "flags": [], "candidateHeadings": [ "Important: If a natural number greater than 1, n, has no prime divisors less than or equal to \\( \\sqrt{n} \\), then n is prime.", "Exercises" ], "candidateMarkers": [ { "id": "2.4.1", "label": null, "offset": 1977 }, { "id": "2.4.2", "label": null, "offset": 2087 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0050.png", "sha256": "6091b071d6a842da9388d9c835dca09cc04bce90908b4734b41bede424b408d8", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 51, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "eb8dfac2625b777f3c5374f29b78f1215082c73967c599aabdc56d3e7fe321ff", "textHash": "6d5b062b8e06b347a7b44cb30d0aba3b538cbbb2dab35c1b48cd22b2814ea50e", "characters": 1865, "flags": [], "candidateHeadings": [ "2.5 Summary", "Definitions:", "Important: A composite number can be written as the product of two natural numbers that are both between 1 and itself:", "Important: 1 is the only natural number that is neither prime nor composite. It has only one positive divisor—itself. This is one more reason why 1 is a very special number.", "Concept: It is far more important to understand how the process of the Sieve works than to remember (or even spell!) Eratosthenes.", "Concept: When faced with a challenging problem, play around with simple examples. They often provide clues that help us discover the solution to the whole problem." ], "candidateMarkers": [ { "id": "2.5", "label": null, "offset": 33 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0051.png", "sha256": "6e9969ef1c109bf730324fa5a468f40c3223c7f863d98d73935c9c3a6b76fea2", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 52, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c962679faca191c4ea50d65da95b47acf797802064f522fb1cb51754189d53aa", "textHash": "80d8c43f1c4bd67e0ee0a34576533eb8a6237f76fc77f2f23f23ef7c296698d3", "characters": 1688, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Concept: When faced with a difficult problem, finding a pattern often helps us guess the solution. A good guess helps us know what to look for when searching for a complete solution.", "Important: If a natural number greater than 1, n, has no prime divisors less than or equal to \\(\\sqrt{n}\\), then n is prime.", "REVIEW PROBLEMS", "2.6 Identify the primes between 200 and 300 by extending the Sieve of Eratosthenes that we began in Section 2.3.", "2.7 How many prime numbers are multiples of 3?", "2.8 How many prime numbers are multiples of 10?", "2.9 Determine which of the following natural numbers are prime and which are composite.", "2.10 Each week, between 30 and 50 students show up for an archery class run by Betty and Wilma. Usually the students break up into groups of equal size for target practice. However, this week, Betty noticed that she could not break the students up into multiple groups of equal size. Wilma noticed that if Betty and Wilma joined the students in practicing, they still could not break the archers up into groups of equal size. How many students showed up to the archery class this week?", "2.11 What is the largest two-digit prime number whose digits are also each prime? (Source: MATHCOUNTS)", "2.12 How many prime numbers are perfect cubes?" ], "candidateMarkers": [ { "id": "2.6", "label": null, "offset": 550 }, { "id": "2.7", "label": null, "offset": 664 }, { "id": "2.8", "label": null, "offset": 712 }, { "id": "2.9", "label": null, "offset": 761 }, { "id": "2.10", "label": null, "offset": 901 }, { "id": "2.11", "label": null, "offset": 1388 }, { "id": "2.12", "label": null, "offset": 1492 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0052.png", "sha256": "19464b60bfe76880ba063df4e68073b2595872a67e8c1e7174aab2d12f6a39f7", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 53, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b4a3a323bdc1025d1e5281fcbbd06669e9f669c315acc8b6575b6a5b8cf383e2", "textHash": "f43e8437b8c466fece1b4743ddf77908b728637821751159f66b6ef066eda788", "characters": 2004, "flags": [], "candidateHeadings": [ "Challenge Problems", "2.13 Find the remainder when the sum of the six smallest primes is divided by the seventh.", "2.15 What is the smallest prime divisor of \\( 5^{23} + 7^{17} \\)?", "2.16 A group of 25 pennies is arranged into three piles such that each pile contains a different prime number of pennies. What is the greatest number of pennies possible in any of the three piles? (Source: MATHCOUNTS) Hints: 120" ], "candidateMarkers": [ { "id": "2.13", "label": null, "offset": 53 }, { "id": "2.14", "label": null, "offset": 145 }, { "id": "2.15", "label": null, "offset": 362 }, { "id": "2.16", "label": null, "offset": 429 }, { "id": "2.17", "label": null, "offset": 659 }, { "id": "2.18", "label": null, "offset": 784 }, { "id": "2.19", "label": null, "offset": 816 }, { "id": "2.20", "label": null, "offset": 891 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/page-0053.png", "sha256": "67cf60c31986491c551b88bef9863fd622eb7d645bff56efed30530da4dc3c1a", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "book-05", "pdfPage": 54, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "497be063fa812caca0098b2fe1662dd9905abed96a7cfbd512cc6838d08cb34e", "textHash": "60fdeb2b50e45f06b605034558cb5646eab325c4da94822ff16b3347dcd3a486", "characters": 930, "flags": [], "candidateHeadings": [ "3.1 Introduction", "3.2 Common Divisors", "Definition: A divisor of every integer in a group is a common divisor of those integers." ], "candidateMarkers": [ { "id": "3.1", "label": null, "offset": 115 }, { "id": "3.2", "label": null, "offset": 368 }, { "id": "3.1", "label": "Problem", "offset": 398 }, { "id": "3.2", "label": "Problem", "offset": 561 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0054.png", "sha256": "5b2607ec96235068fa289c921f5271e5c77d253a30439f05c8f0d1789cec568a", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 55, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d6d10417cee8017842fa9d72e11de7cb2dbe570a8014e3ea5112b7e9341b60b9", "textHash": "0b260217696af75222b6f3c016313d97df7b4707d9507df5def46861d2e4d69a", "characters": 2130, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "3.1", "label": "Problem", "offset": 265 }, { "id": "3.1", "label": "Solution for Problem", "offset": 334 }, { "id": "3.2", "label": "Problem", "offset": 682 }, { "id": "3.2", "label": "Solution for Problem", "offset": 959 }, { "id": "3.2.1", "label": null, "offset": 1542 }, { "id": "3.2.2", "label": null, "offset": 1601 }, { "id": "3.2.3", "label": null, "offset": 1660 }, { "id": "3.2.4", "label": null, "offset": 1719 }, { "id": "3.2.5", "label": null, "offset": 1778 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0055.png", "sha256": "d84b3b801bd634000ac283e47f191e7b60402d3dce831610778ef2991780471a", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 56, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "08736484ae620b1332248d280cca6f0565cf18e2f80dbc6198287c1994c2aff2", "textHash": "11bdf3553b698fdd1f891f92c6713ea05fb133af21004044f8edd08e4ba80ffe", "characters": 1601, "flags": [], "candidateHeadings": [ "3.3 Greatest Common Divisors (GCDs)", "Definition: Of the common divisors of a group of integers, we call the largest one the greatest common divisor (GCD).", "Important: Some sources refer to the GCD of a group of integers as the greatest common factor (GCF) of the group of integers. Just remember that these refer to the same thing!" ], "candidateMarkers": [ { "id": "3.2.6", "label": null, "offset": 37 }, { "id": "3.3", "label": null, "offset": 383 }, { "id": "3.3", "label": "Problem", "offset": 430 }, { "id": "3.4", "label": "Problem", "offset": 506 }, { "id": "3.5", "label": "Problem", "offset": 698 }, { "id": "3.3", "label": "Problem", "offset": 1335 }, { "id": "3.3", "label": "Solution for Problem", "offset": 1376 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0056.png", "sha256": "4da0936da914da39394ec911901738c96db550580f7fbd0abdc96118f2613a9f", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 57, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ff678c4cf9a29b9b495ff43e4dc7b9b6f5cfbb4f2d0804f1a51efb6a2ade0483", "textHash": "0967f136e446561e411d94132e740079552d77c3172811c0783b3da32e9ddf7c", "characters": 1676, "flags": [], "candidateHeadings": [ "Important: There is a special notation for the GCD of two integers. We can write the GCD of 12 and 18 as gcd(12, 18). For instance, we write", "Definition: When the only common positive divisor of a pair of integers is 1, we say that those integers are relatively prime. A pair of relatively prime integers is sometimes referred to as coprime, which means the same thing." ], "candidateMarkers": [ { "id": "3.4", "label": "Problem", "offset": 899 }, { "id": "3.4", "label": "Solution for Problem", "offset": 991 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0057.png", "sha256": "89cbcafdecf70e3052caed4aeb5bc838f8adebc2a32e9e0bb7d8be9eb46b3c4f", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 58, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2a2dbd346556f007ed94308a33c91c04154788d08d8cd4b659ece30a2939c8c2", "textHash": "cffb05123bcf335503dd1a81c006733321ad6ebba753db91d8de53f4894fa3c2", "characters": 2026, "flags": [], "candidateHeadings": [ "Exercises", "3.4 Common Multiples", "Definition: A multiple of every integer in a group is a common multiple of those integers." ], "candidateMarkers": [ { "id": "3.5", "label": "Problem", "offset": 22 }, { "id": "3.5", "label": "Solution for Problem", "offset": 241 }, { "id": "3.3.1", "label": null, "offset": 787 }, { "id": "3.3.2", "label": null, "offset": 968 }, { "id": "3.4", "label": null, "offset": 1234 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0058.png", "sha256": "3b255afee25f86a559700e9e13d6e7ac19b5e8fd18026d5b83e1e472073ab2d7", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 59, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8e0c83078a398bf30698399d7d3f93a8dd9a95527d8f8ad14f895008a4c591ac", "textHash": "d7f68b9eee3bf2ccb0740a15181fa1cc4edd886af5646e1c372da069075f59d5", "characters": 2024, "flags": [], "candidateHeadings": [ "Definition: The smallest positive common multiple of a group of integers is the least common multiple (LCM) of those integers." ], "candidateMarkers": [ { "id": "3.6", "label": "Problem", "offset": 540 }, { "id": "3.7", "label": "Problem", "offset": 624 }, { "id": "3.8", "label": "Problem", "offset": 664 }, { "id": "3.6", "label": "Problem", "offset": 709 }, { "id": "3.6", "label": "Solution for Problem", "offset": 793 }, { "id": "3.7", "label": "Problem", "offset": 1212 }, { "id": "3.7", "label": "Solution for Problem", "offset": 1252 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0059.png", "sha256": "d003e26772ec7772e7ed1fedd056ceaf98d1f1c5c525db7696d8512365ce8274", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 60, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "186880bd072855fedfc2b57d198f6e79f7ecb88a7b6a582aafb47b0ab1b3d149", "textHash": "f931d8c730ad7813b05e025941653a1cf68bc1f5f2d08e92dfe17d90e3a003cb", "characters": 1846, "flags": [], "candidateHeadings": [ "Important: There is a special notation for the LCM of two integers. We can write the LCM of 12 and 18 as lcm[12, 18]. For instance,", "Exercises" ], "candidateMarkers": [ { "id": "3.8", "label": "Problem", "offset": 22 }, { "id": "3.8", "label": "Solution for Problem", "offset": 67 }, { "id": "3.4.1", "label": null, "offset": 1328 }, { "id": "3.4.2", "label": null, "offset": 1696 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0060.png", "sha256": "f090b4cc924b6afd2964016823f01e5571026a0b97bb2504d7a7a63265337202", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 61, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "110a480a9bb4c25ca73251357d6cac1c66c815e61fae773e50bdf32fbde1b877", "textHash": "160e627eab9e5902e38d918e5e96d949e63af6030e06368ba4ccf5931ee74743", "characters": 7438, "flags": [ "truncated" ], "candidateHeadings": [ "3.5 Remainders" ], "candidateMarkers": [ { "id": "3.5", "label": null, "offset": 34 }, { "id": "3.9", "label": "Problem", "offset": 50 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0061.png", "sha256": "2e7d34916c8bcdf044f4005d8b70777002d824229599a413adc1c9497fb8df84", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 62, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c5358e4686fc152fbc93a7a9f3275f45d20f66bc44e5274fd61c1e6329f6608b", "textHash": "99c8e73116dd93866b9de7d6955003651c8f552c9be7234c7fafba0feedabbaf", "characters": 17735, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "3.9", "label": "Problem", "offset": 79 }, { "id": "3.9", "label": "Solution for Problem", "offset": 597 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0062.png", "sha256": "fc94cc2b97ed8142ef6571d625c8c73ab4c338dc019fc76898e259e7ae76c091", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 63, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c622eb8a5fecd26f22c2977c1cd7e8c14c1dcdcc5f2b96def45075bcad65ccb1", "textHash": "ef8543b9cc3b1312fd1cff7b1f446ac7f0212f6fd9de43fcd99089f6921cba36", "characters": 2310, "flags": [], "candidateHeadings": [ "Important: The Division Theorem says that for any integer \\(a\\) and positive integer \\(b\\), there exists exactly one pair of integers \\(q\\) and \\(r\\) such that", "Definitions: In the Division Theorem, we call \\(a\\) the dividend, \\(b\\) the divisor, \\(q\\) the quotient, and \\(r\\) the remainder. (Note that the term divisor is used differently everywhere else in the book.)" ], "candidateMarkers": [ { "id": "3.9", "label": "Problem", "offset": 34 }, { "id": "3.10", "label": "Problem", "offset": 878 }, { "id": "3.10", "label": "Solution for Problem", "offset": 979 }, { "id": "3.11", "label": "Problem", "offset": 1622 }, { "id": "3.11", "label": "Solution for Problem", "offset": 1723 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0063.png", "sha256": "a2a966273677884e26bd0b359303ca2c4ab95387250cbad32a47ace93b53f657", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 64, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "01fbfdcb87ee9cec3ff7d02b3ce085851fba807544a29527d9eeef504bbabe64", "textHash": "bf63abfa4437d299d2bd8bbeec2507472baa5a82ccdc9778ca788687f8b2d870", "characters": 1569, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "3.12", "label": "Problem", "offset": 332 }, { "id": "3.12", "label": "Solution for Problem", "offset": 435 }, { "id": "3.13", "label": "Problem", "offset": 1158 }, { "id": "3.13", "label": "Solution for Problem", "offset": 1263 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0064.png", "sha256": "c34795ba9d554603aaf3cb65e03258605f7198049bcabbdbdeee931af2c8c031", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 65, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6ac3a467551456a529870dd36621e403355baa510d39e68f0d61a42b1ed9d626", "textHash": "f5329d47fc1eea5b9a10562d22c966e49ad62f9391f75e1e1054938c6f2a7f5f", "characters": 1937, "flags": [], "candidateHeadings": [ "Exercises", "3.6 Multiples, Divisors, and Arithmetic" ], "candidateMarkers": [ { "id": "3.5.1", "label": null, "offset": 240 }, { "id": "3.5.2", "label": null, "offset": 333 }, { "id": "3.5.3", "label": null, "offset": 426 }, { "id": "3.5.4", "label": null, "offset": 503 }, { "id": "3.5.5", "label": null, "offset": 582 }, { "id": "3.5.6", "label": null, "offset": 669 }, { "id": "3.6", "label": null, "offset": 748 }, { "id": "3.14", "label": "Problem", "offset": 799 }, { "id": "3.15", "label": "Problem", "offset": 1296 }, { "id": "3.16", "label": "Problem", "offset": 1406 }, { "id": "3.17", "label": "Problem", "offset": 1812 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0065.png", "sha256": "36ddf87992e165e7867aaff27ef7276347f5333743e4adeef6cea6ba98ef7c83", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 66, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "36efa6516833f0fd7a8ffa367e43bc901ee5655ae1d9d7f3542be6e9b99e1059", "textHash": "96e5082dd578f3aa23e013c10b1b56dd13ce06b5f6526d0fd8c2cddf0ec5e246", "characters": 2147, "flags": [], "candidateHeadings": [ "Concept: Solving a problem often means finding a new, more helpful way to look at it. Expressing integers as products of other integers helps with many number theory problems by giving us another, more helpful view of those integers." ], "candidateMarkers": [ { "id": "3.14", "label": "Problem", "offset": 41 }, { "id": "3.14", "label": "Solution for Problem", "offset": 127 }, { "id": "3.15", "label": "Problem", "offset": 1666 }, { "id": "3.15", "label": "Solution for Problem", "offset": 1776 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0066.png", "sha256": "941602e0bc3c8496335d1c78b0bcc18ce773644a57f660f5c652348e05d57834", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 67, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "18c204e3d0fcfc11499105119d04f38adb26cbc3c8b07cb9adeec36869920e19", "textHash": "46839abc7df4a10879a0e384e22cf9ab5b4993d18e2f1e7a01674835e8fb5945", "characters": 2535, "flags": [], "candidateHeadings": [ "Important:" ], "candidateMarkers": [ { "id": "3.16", "label": "Problem", "offset": 406 }, { "id": "3.16", "label": "Solution for Problem", "offset": 504 }, { "id": "3.17", "label": "Problem", "offset": 1376 }, { "id": "3.17", "label": "Solution for Problem", "offset": 1498 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0067.png", "sha256": "1feac31bb08c684277826a77b4d1b354d5f78211ae6cba73ae9d2c3b10c9c9ba", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 68, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2134b40f7ea61092f1aec0d1b94eb8729bda1ad86134356dc0e2a211525f7ff5", "textHash": "369648b47555cce214d649bf515bdbf4c55dd78da9e48734124028fb421915dd", "characters": 1970, "flags": [], "candidateHeadings": [ "Exercises", "3.7 The Euclidean Algorithm" ], "candidateMarkers": [ { "id": "3.6.1", "label": null, "offset": 40 }, { "id": "3.6.2", "label": null, "offset": 534 }, { "id": "3.6.3", "label": null, "offset": 1108 }, { "id": "3.6.4", "label": null, "offset": 1211 }, { "id": "3.6.5", "label": null, "offset": 1316 }, { "id": "3.6.6", "label": null, "offset": 1390 }, { "id": "3.7", "label": null, "offset": 1484 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0068.png", "sha256": "5eb8059457da4e1d6660450ee84c07bddc04daf7023a26ea21a1cd57044a4d4f", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 69, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c7b8adf39126d3be02332715fe9cda7de2d34422abea2f540980e0a8ea617c7b", "textHash": "f0a2db6e488be6e112217d9778893410a7d9b6bd139d74347eeef1cf9475a97b", "characters": 1940, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "3.18", "label": "Problem", "offset": 44 }, { "id": "3.19", "label": "Problem", "offset": 464 }, { "id": "3.20", "label": "Problem", "offset": 566 }, { "id": "3.21", "label": "Problem", "offset": 781 }, { "id": "3.22", "label": "Problem", "offset": 816 }, { "id": "3.18", "label": "Problem", "offset": 1050 }, { "id": "3.18", "label": "Solution for Problem", "offset": 1157 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0069.png", "sha256": "535e8c567000b0440f00324682b3b8a9c4d791c7bdb54b2fa94d8b6b6da695dd", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 70, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "683824dd24cb099fb7de051a49dc179bc5e20e69541741c1b630580c4d2ccb1c", "textHash": "1bfeba199520d06caa486241bb96653b364a61eed7c97f4422b8349c0d2e51a2", "characters": 2444, "flags": [], "candidateHeadings": [ "Concept: Knowing why the mathematics we use works gives us a deeper understanding of the math. We can then apply our understanding to difficult problems." ], "candidateMarkers": [ { "id": "3.18", "label": "Problem", "offset": 1277 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0070.png", "sha256": "5bddd949affba675a059278836936b78a6f0535391f7f7216064c7d491a38a04", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 71, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f395109df06d211c30452706b97eedea6850729f50cc57319b06c4e2ce5eba98", "textHash": "9e49222301501d75f7200a491f47362b2374bdaea3dd36edad5b7256bad09fd4", "characters": 2141, "flags": [], "candidateHeadings": [ "Important: For natural numbers m and n, where m > n,", "Concept: The GCD of two different natural numbers is the same as the GCD of the smaller of the two natural numbers and their difference. This is helpful because it is almost always simpler to find the divisors of smaller integers." ], "candidateMarkers": [ { "id": "3.19", "label": "Problem", "offset": 34 }, { "id": "3.19", "label": "Solution for Problem", "offset": 143 }, { "id": "3.20", "label": "Problem", "offset": 1810 }, { "id": "3.20", "label": "Solution for Problem", "offset": 1850 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0071.png", "sha256": "6dad573d466c0a4825639a290f93ecb520851f68c6003775fb8393f9963009a0", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 72, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "55f77e14f83b1d138b73fc37a942b2b4bfc4418cc0e513e12bae9d2750191e9e", "textHash": "b4af88dd8bf9f4b9bc58933791ec3747dbd5e87c5381bcb7281ed559a04894f3", "characters": 1381, "flags": [], "candidateHeadings": [ "Important: This method for finding the GCD comes from repeatedly using the fact that" ], "candidateMarkers": [ { "id": "3.21", "label": "Problem", "offset": 1126 }, { "id": "3.21", "label": "Solution for Problem", "offset": 1161 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0072.png", "sha256": "39edcc0771f63ecd870d7d79650202e81c36e0fc05936622c226031d3f5d71d9", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 73, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8e0a2d8b00cb951adb0a1062c83ad871730625fb6a47388dd2d6f2e3a4cd7ec0", "textHash": "4d3cae427f8560036846570606c3a4fa0c896629c56a6204dd31013ed3c0dc11", "characters": 2251, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "3.22", "label": "Problem", "offset": 544 }, { "id": "3.22", "label": "Solution for Problem", "offset": 778 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0073.png", "sha256": "75d035dc33d2c712d33b0ccdb2fad355d9230c6382ad9ee48900b5a168119201", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 74, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "125a8cf5363ffac69c4a1c54bf515e0d61fffc76aacc4f4d335afaa10e1858f6", "textHash": "d3b27f88321df2e21eae12770c8c99e4d90666b53fb23ce8d99f6826b4e3cb8a", "characters": 1500, "flags": [], "candidateHeadings": [ "Important: Let m and n be integers such that m = qn + r, where \\(0 \\leq r < n\\). Then" ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0074.png", "sha256": "f8e829dfef552ae38bbae83c07299206ff8a62ff676cc886fabaedef3de8006d", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 75, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7234062e04d1c72aea8f3be114640c896ea9a20fd86ae8d881bd29e03df7cffa", "textHash": "359a25df88c68886723c59d3d37d722583aa8eeae3516ae600b7a3129832258b", "characters": 1691, "flags": [], "candidateHeadings": [ "Exercises", "3.8 Summary", "Definitions:", "Important:", "Concept:", "Important:", "Important:" ], "candidateMarkers": [ { "id": "3.7.1", "label": null, "offset": 45 }, { "id": "3.8", "label": null, "offset": 238 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0075.png", "sha256": "97a4acdd42973a7b6c19100844cd018330fbafa408b681c87890d9210aa9d493", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 76, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2857d2303ea2e4b62dfdc9e7f7f1d3c9fd0daaed7b484a8ee8ac7a889912953f", "textHash": "bced635eec9cfba4f1e8363e5422a3b5f4ca08c28b2d098399f71f8064eb29fd", "characters": 1396, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Important:", "REVIEW PROBLEMS", "3.23 Find each of the following.", "3.24 List the positive common divisors of each of the following groups of integers.", "3.25 Find each of the following.", "3.26 What is the least common multiple of 12, 18, and 30? (Source: MATHCOUNTS)", "3.27 What is the smallest number of marbles that could either be divided up into bags of 18 marbles or into bags of 42 marbles with no marbles leftover in each case?", "3.28 What is the result when we decrease the greatest common divisor of 6432 and 132 by 8? (Source: AMC)", "3.29 How many of the first 100 positive integers are divisible by 7? (Source: MATHCOUNTS)" ], "candidateMarkers": [ { "id": "3.23", "label": null, "offset": 322 }, { "id": "3.24", "label": null, "offset": 520 }, { "id": "3.25", "label": null, "offset": 662 }, { "id": "3.26", "label": null, "offset": 949 }, { "id": "3.27", "label": null, "offset": 1029 }, { "id": "3.28", "label": null, "offset": 1196 }, { "id": "3.29", "label": null, "offset": 1302 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0076.png", "sha256": "a3951368e00b0dcb4eb6508e73c1635ec1bc3bf4f8e632d0074b1fab6fb7db83", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 77, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fc912a9e8f8726dd29bafb13b34109768b90cf7e4783b65acac0883c727ee791", "textHash": "481f9163f3135c87ebc0d5ccf21b9d65cef2e9792fa533d05ea3396c07d03315", "characters": 1651, "flags": [], "candidateHeadings": [ "Challenge Problems", "3.30 Mr. Namm baked 252 cookies, Mrs. Clancy baked 105 cookies, and Mr. Palavas baked 168 cookies. Each baker packaged them with the same number of cookies in each package. What is the greatest number of cookies that could be in each package? (Source: MATHCOUNTS) Hints: 73", "3.31 We can construct 5 and 10 as differences of multiples of 25 and 15:", "2.25 - 3.15 = 5", "1.25 - 1.15 = 10", "3.32 There are 25 primes less than 100. Is their sum even or odd? Hints: 23", "3.34 Tom, Dick, and Harry collect coins. While looking through their combined collection of silver dollars, Tom (who never miscounts) mentions that exactly a third of the coins are his, exactly a fourth of the coins are Dick's, and rest belong to Harry. Without counting the coins, Harry correctly declares, \"the total number of coins is a multiple of 12.\" Dick adds, \"Harry's total number of coins is a multiple of 5.\" Explain how Dick and Harry knew this without counting the coins themselves. Hints: 66", "3.35 What is the smallest positive four-digit number that gives a quotient 432 with remainder 2 when divided by a positive one-digit number? (Source: MATHCOUNTS)" ], "candidateMarkers": [ { "id": "3.30", "label": null, "offset": 54 }, { "id": "3.31", "label": null, "offset": 329 }, { "id": "2.25", "label": null, "offset": 403 }, { "id": "1.25", "label": null, "offset": 420 }, { "id": "3.32", "label": null, "offset": 507 }, { "id": "3.33", "label": null, "offset": 584 }, { "id": "3.34", "label": null, "offset": 717 }, { "id": "3.35", "label": null, "offset": 1224 }, { "id": "3.36", "label": null, "offset": 1387 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/page-0077.png", "sha256": "ab58c245cf90e8bda4fc834319c5e1de630684bffcf822fa27bb0579f225e5ea", "nextChapterHeadingPDF": 78 } }, { "source": "book-05", "pdfPage": 78, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "33e7f7ea29fa36d79f0eef34f1f0e4fe166b86fb40c4a1ae68622339aeeaf694", "textHash": "2850798e6e0463a016b26d8e099965376fd408f2efaa7d99e8f82f131a026c49", "characters": 1363, "flags": [], "candidateHeadings": [ "4.1 Introduction" ], "candidateMarkers": [ { "id": "4.1", "label": null, "offset": 144 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0078.png", "sha256": "7a39a8f36c66255448abc35998d4ad0da4e989bd6a3a675648dd841577abd05f", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 79, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a004e22faa991a9c6de98c85d0fafb1e094bc29f3dd679d44e4b0e0885fa2d7d", "textHash": "655bc41be210c5c62c13d16665d28875fa680aed36d656cede22b339809fc182", "characters": 1249, "flags": [], "candidateHeadings": [ "4.2 Factor Trees" ], "candidateMarkers": [ { "id": "4.2", "label": null, "offset": 31 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0079.png", "sha256": "393b8eca8a225972949294ac1e95248e8be6a432a0963c8efbdd668d0e366b32", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 80, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7c9293e40285ad04968a07abfe3cbcaab1e3c6ed54a02c5ac38d8b0a2bbe9457", "textHash": "879ce7cf3cb8dd88160a7aadf162ed4fa8f5bdd0ea6b34a679a4578c873f37d6", "characters": 1825, "flags": [], "candidateHeadings": [ "Definition: A factor tree is a visual chart that shows a step-by-step process of breaking an integer down into a product of primes." ], "candidateMarkers": [ { "id": "4.1", "label": "Problem", "offset": 433 }, { "id": "4.2", "label": "Problem", "offset": 501 }, { "id": "4.3", "label": "Problem", "offset": 569 }, { "id": "4.4", "label": "Problem", "offset": 637 }, { "id": "4.5", "label": "Problem", "offset": 706 }, { "id": "4.6", "label": "Problem", "offset": 775 }, { "id": "4.1", "label": "Problem", "offset": 1447 }, { "id": "4.1", "label": "Solution for Problem", "offset": 1508 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0080.png", "sha256": "fd1f59614c8ba8aa486b587cda358d16685d7e886db0f7206aacd79fba2c0f8d", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 81, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7690a61b9b19c95fd2a82fb31b9e1a831e8366493862c89419b06b5f475eefd5", "textHash": "664526440c72b566f1a7d009a673c3cc098d570d19066306d8fbae072259d207", "characters": 1658, "flags": [], "candidateHeadings": [ "Definition: The prime factorization of an integer is the product of primes equal to that integer." ], "candidateMarkers": [ { "id": "4.2", "label": "Problem", "offset": 1185 }, { "id": "4.2", "label": "Solution for Problem", "offset": 1235 }, { "id": "4.3", "label": "Problem", "offset": 1550 }, { "id": "4.3", "label": "Solution for Problem", "offset": 1600 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0081.png", "sha256": "6d12f300459dc4cebb3d8d6393f4fa017bdd2dabbbb61d2fde0b81c8f55fabc3", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 82, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3dcc0ab3afa46cb727dfe9c8180bd16395cfd51ce4ac3e02655393dab4a8c026", "textHash": "678853cdfe74267ae454fd53bc586c3db06632d28d05a45ccbf991a7048f5618", "characters": 1157, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "4.4", "label": "Problem", "offset": 18 }, { "id": "4.4", "label": "Solution for Problem", "offset": 69 }, { "id": "4.5", "label": "Problem", "offset": 125 }, { "id": "4.5", "label": "Solution for Problem", "offset": 176 }, { "id": "4.6", "label": "Problem", "offset": 801 }, { "id": "4.6", "label": "Solution for Problem", "offset": 891 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0082.png", "sha256": "1edd3b4f2d5f7e4c449aff4d4889fd86465887bf0ae8d1dc05731c843a811e06", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 83, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "91c3228899921f09788197cb62ac15e21ae7b79848fa26243c772d39d8fcc1bb", "textHash": "7c943be4f79ff7b6079b11bf78ed6914bcc22e4d4a66fba3601cce5957bf9a21", "characters": 1783, "flags": [], "candidateHeadings": [ "Concept: We now see one reason why prime numbers are so important: all composites numbers can be factored into products of only prime numbers. Nearly every concept in this book from this point on depends in some way on prime factorization. Prime factorization is therefore a cornerstone of number theory.", "Important: Every positive integer has exactly one prime factorization. This fact is known as the Fundamental Theorem of Arithmetic.", "Exercises", "4.3 Factorization and Multiples" ], "candidateMarkers": [ { "id": "4.2.1", "label": null, "offset": 1192 }, { "id": "4.3", "label": null, "offset": 1380 }, { "id": "4.7", "label": "Problem", "offset": 1423 }, { "id": "4.8", "label": "Problem", "offset": 1703 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0083.png", "sha256": "e37c6eface2aeb6309a21ad1545ca59915f3ba0d4a540f480dbc3020c2049a6a", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 84, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e62783df5b05740ec861adacc33bc57a842b45beb6f9d6d03ab8ee72e0e48e48", "textHash": "6a5ba4f859d6e6015a840f6e37fed7576b739c61a5bc1e0a9422320978edf4e9", "characters": 1761, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "4.9", "label": "Problem", "offset": 33 }, { "id": "4.10", "label": "Problem", "offset": 212 }, { "id": "4.7", "label": "Problem", "offset": 479 }, { "id": "4.7", "label": "Solution for Problem", "offset": 579 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0084.png", "sha256": "fd23681a56a0d7f785997b989893ef26fa633f84e88a48b65f8570a1ece81285", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 85, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b3938fd09f3f30172857819cd09d01b8528e8b7faba2116414a114de87026891", "textHash": "e1df4d99d9e649d34bfa6657853c609a499699f28ad6e47e598621097a5d9105", "characters": 2048, "flags": [], "candidateHeadings": [ "Important: The prime factorization of any multiple of a natural number \\(m\\) contains the entire prime factorization of \\(m\\).", "Concept: Prime factorizations help us organize important information about integers that becomes useful in solving problems more quickly. We can then find solutions to problems by constructing the prime factorizations of the solutions.", "Important: The primes in the prime factorization of the LCM of a group of integers are those primes present in any of the prime factorizations of the numbers in the group. The exponent of each such prime in the LCM's prime factorization is the greatest exponent of that prime present among the prime factorizations of the numbers in the group." ], "candidateMarkers": [ { "id": "4.8", "label": "Problem", "offset": 159 }, { "id": "4.8", "label": "Solution for Problem", "offset": 195 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0085.png", "sha256": "8b24b616323a1029e87186988278bcdd495d9e98ec7ef9213dde96e9d629ca11", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 86, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c6e6095c173576b0bd7e68eacdd10c2e00e574607a2005aee6a8a132a3ca56b0", "textHash": "94cc417ca76685fd1264b86ed5f426bb573944803c9fc4a7aaa9ae9ac268a2b6", "characters": 1828, "flags": [], "candidateHeadings": [ "Important: Let a and b be two positive integers. Let n be the number of prime factors of a or b. Let \\(p_1, p_2, \\ldots, p_n\\) be those prime factors. Now we write a and b in terms of those prime factors:" ], "candidateMarkers": [ { "id": "4.9", "label": "Problem", "offset": 856 }, { "id": "4.9", "label": "Solution for Problem", "offset": 931 }, { "id": "4.10", "label": "Problem", "offset": 1600 }, { "id": "4.10", "label": "Solution for Problem", "offset": 1685 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0086.png", "sha256": "5a8145d97a2f7db34997ad62495ca8b41408c7dffd027415a1e6417085f80e98", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 87, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b93b253f4a3080c27668d420b853e8988e1b8ac43b8e31119ab6ce87c9808cd0", "textHash": "c77e5c5c8bf4e6b4cc75180df47e946cda9d5ae6ea3e118ff9208cdf5627201e", "characters": 1424, "flags": [], "candidateHeadings": [ "Exercises", "4.4 Factorization and Divisors" ], "candidateMarkers": [ { "id": "4.3.1", "label": null, "offset": 192 }, { "id": "4.3.2", "label": null, "offset": 470 }, { "id": "4.3.3", "label": null, "offset": 537 }, { "id": "4.3.4", "label": null, "offset": 609 }, { "id": "4.4", "label": null, "offset": 705 }, { "id": "4.11", "label": "Problem", "offset": 747 }, { "id": "4.12", "label": "Problem", "offset": 1014 }, { "id": "4.13", "label": "Problem", "offset": 1097 }, { "id": "4.11", "label": "Problem", "offset": 1190 }, { "id": "4.11", "label": "Solution for Problem", "offset": 1300 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0087.png", "sha256": "c3df35fca45f2fe41fccbb7bb81760be5b5d1a012123dc88c98a21f3cffe7ee8", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 88, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "71f8eca32f2f530e36d7b68055ce950c5f79fc9258a30fa2fd39214ac4ed19bd", "textHash": "cb38415af609f7aad16c1dfebba61ca2d13f233e2d3983b584a98d1408af0bd8", "characters": 1363, "flags": [], "candidateHeadings": [ "Important: The primes in the prime factorization of the GCD of a group of integers are those primes present in all of the prime factorizations of the numbers in the group. The exponent of each prime in the GCD's prime factorization is the smallest exponent of that prime present among the prime factorizations of the numbers in the group." ], "candidateMarkers": [ { "id": "4.12", "label": "Problem", "offset": 32 }, { "id": "4.12", "label": "Solution for Problem", "offset": 65 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0088.png", "sha256": "5e84e83cbb195c47a5e61a3d7be0e0036113d8a82994d9e3731d29d48778ed18", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 89, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "376cb683f848a2796895e9f3e422039801946f6d29c6bce764dc5d128f913010", "textHash": "555508a8a8abd035f5d0c27539ef376cd264565bbec921b2a7b3c978a66ef62d", "characters": 1901, "flags": [], "candidateHeadings": [ "Important: Let a and b be two positive integers. Let n be the number of prime factors of a or b. Let \\(p_1, p_2, \\ldots, p_n\\) be those prime factors. Now we write a and b in terms of those prime factors:" ], "candidateMarkers": [ { "id": "4.13", "label": "Problem", "offset": 943 }, { "id": "4.13", "label": "Solution for Problem", "offset": 1047 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0089.png", "sha256": "e4decedef7460d0a984f58a733669486d52456c61b0269b1896f5e387c201231", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 90, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c57d726fd4108cef75ca34527d2a696c85e6c61abfd236a5e24fdcd6265303d5", "textHash": "967d2619833c932e0bc1197b91fecd4c67f2a7390cf354cd97504dab4d657e8b", "characters": 1524, "flags": [], "candidateHeadings": [ "Exercises", "4.5 Rational Numbers and Lowest Terms", "Definition: Numbers that can be expressed as the quotient of two integers are called rational numbers." ], "candidateMarkers": [ { "id": "4.4.1", "label": null, "offset": 50 }, { "id": "4.4.2", "label": null, "offset": 220 }, { "id": "4.5", "label": null, "offset": 364 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0090.png", "sha256": "a8e7f39f26d98c86a03307393bf1135282359333cac3af8d811e8364f5c02a67", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 91, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4486e7ec53f96e436084256ef7cb3fb84e7f662ab7af5cc99ec5a95984441029", "textHash": "528ae7000f3e6d8ac5ba79ebd3e424968b26c076531c6057c149ed4566251e3d", "characters": 1333, "flags": [], "candidateHeadings": [ "Definition: When the numerator and denominator of a fraction are relatively prime, we say that the fraction is written in lowest terms or as a fraction in reduced form. When the GCD of the numerator and denominator of a fraction is greater than 1, the fraction is in unreduced form." ], "candidateMarkers": [ { "id": "4.14", "label": "Problem", "offset": 903 }, { "id": "4.15", "label": "Problem", "offset": 1083 }, { "id": "4.14", "label": "Problem", "offset": 1159 }, { "id": "4.14", "label": "Solution for Problem", "offset": 1218 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0091.png", "sha256": "e64e7adddcd3434a5d5cdaa142af987e26eac95bf582779fd33865a9e3d19c19", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 92, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7409695e4afbef01e991cea2b448f8cb39d8ad6ace47934b12fba1b431d8bb64", "textHash": "a4b0e8c1bcc6fc49d39909a54d6f4f6f157dda3f20e03a1534dd2397fcda7681", "characters": 9737, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [ "Important: Prime factorizations help us quickly reduce fractions to lowest terms.", "Exercises" ], "candidateMarkers": [ { "id": "4.15", "label": "Problem", "offset": 607 }, { "id": "4.15", "label": "Solution for Problem", "offset": 683 }, { "id": "4.5.1", "label": null, "offset": 1218 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0092.png", "sha256": "5b01d039dd9644270b6133ee8169824dad8f799776391a446377098757a8e246", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 93, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4fbfe9b10a3bde85064ef5c7b36963483272595e317d4d994149fe6e703d96e7", "textHash": "dc260e79b8410d5a3edafaa280503a5f0554f8a7486ae8a14c47fe16f56d7fd6", "characters": 1761, "flags": [], "candidateHeadings": [ "4.6 Prime Factorization and Problem Solving", "Important: We often use the properties of different kinds of integers to learn about their prime factorizations." ], "candidateMarkers": [ { "id": "4.6", "label": null, "offset": 31 }, { "id": "4.16", "label": "Problem", "offset": 169 }, { "id": "4.17", "label": "Problem", "offset": 359 }, { "id": "4.18", "label": "Problem", "offset": 453 }, { "id": "4.19", "label": "Problem", "offset": 650 }, { "id": "4.16", "label": "Problem", "offset": 889 }, { "id": "4.16", "label": "Solution for Problem", "offset": 976 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0093.png", "sha256": "a706ba87336434367eecfd0117c8be5d439bc43ef5277c9423b166ebb83f7a8e", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 94, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e8d74a3ed5324e88f8534a3cb52c0719e0a6dcd94dd1acd21c3fce354758a2cb", "textHash": "b29775cfc521a0b408d6e4131b82c5a7051908f718f4725ff4b68a846e2c0499", "characters": 2222, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "4.17", "label": "Problem", "offset": 45 }, { "id": "4.17", "label": "Solution for Problem", "offset": 139 }, { "id": "4.18", "label": "Problem", "offset": 598 }, { "id": "4.18", "label": "Solution for Problem", "offset": 686 }, { "id": "4.19", "label": "Problem", "offset": 1187 }, { "id": "4.19", "label": "Solution for Problem", "offset": 1426 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0094.png", "sha256": "d5ac2764d8d925efc796c2123cb23d4924f93c9af2107cee46dbb1a4f46a70ab", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 95, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f78bdd77bf66a1e8a67d579fd3a45d9972fc7b652001d6c772b5679d467fcd82", "textHash": "71f7274ecd4ade405f59624de42e4df9d321287e062720ec957e242e834f2b3d", "characters": 1740, "flags": [], "candidateHeadings": [ "Exercises", "4.7 Relationships Between LCMs and GCDs" ], "candidateMarkers": [ { "id": "4.6.1", "label": null, "offset": 318 }, { "id": "4.6.2", "label": null, "offset": 398 }, { "id": "4.6.3", "label": null, "offset": 482 }, { "id": "4.6.4", "label": null, "offset": 578 }, { "id": "4.6.5", "label": null, "offset": 657 }, { "id": "4.7", "label": null, "offset": 891 }, { "id": "4.20", "label": "Problem", "offset": 942 }, { "id": "4.21", "label": "Problem", "offset": 1278 }, { "id": "4.22", "label": "Problem", "offset": 1333 }, { "id": "4.23", "label": "Problem", "offset": 1640 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0095.png", "sha256": "373c293e43fa1ce68d579c8997f60f9e4f448f39e250237a7bb9cc0e23ad1826", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 96, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6405e2bbe738d99529b66953ccee6203715226bb9aebf903cefbdc38422d0f0b", "textHash": "b6f84c6efdb7ed3b6edd708828b6fec1e1aadeb1500d967c7e58b014dc1c5d25", "characters": 2034, "flags": [], "candidateHeadings": [ "80.2 = 160 = 25.30.51 = 21. (24.30.51)" ], "candidateMarkers": [ { "id": "4.20", "label": "Problem", "offset": 368 }, { "id": "4.20", "label": "Solution for Problem", "offset": 450 }, { "id": "80.2", "label": null, "offset": 784 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0096.png", "sha256": "cbf342556d94a62414ef834f8e6d75c3188ed7cbd3591fa7f3713d46dd03cd35", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 97, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d2c23f123de113cd103ba2e9324eef3329b43c8aa867f78833f0fcec7275ed58", "textHash": "35ba88d7ca54897c8ba124cfb91ab7e6f62508d309da26c3d50e40a24f011904", "characters": 1900, "flags": [], "candidateHeadings": [ "80.3 = 240 = 24.31.51 = 31.24.30.51", "80.4 = 320 = 26.30.51 = 22.24.30.51", "80.5 = 400 = 24.30.52 = 51.24.30.51", "80.6 = 480 = 25.31.51 = 21.31.24.30.51" ], "candidateMarkers": [ { "id": "80.3", "label": null, "offset": 159 }, { "id": "80.4", "label": null, "offset": 362 }, { "id": "80.5", "label": null, "offset": 580 }, { "id": "80.6", "label": null, "offset": 798 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0097.png", "sha256": "30560e919d94fe750f50ffdf8a4e5a89d44d36b78adeb7ab84ac659e3b1f21c5", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 98, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4d5ce15feb6e781a2cf4c337fe75a510cff90ffb30706973381d842085dec946", "textHash": "75ed9a4f6ffe858a59673e062846e2e280ce570258a39f43036ade179e6e4ed2", "characters": 2198, "flags": [], "candidateHeadings": [ "Important: For any positive integers \\(a\\), \\(b\\), and \\(c\\),", "Important: The product of two natural numbers, \\(m\\) and \\(n\\), is equal to the product of their GCD and LCM:" ], "candidateMarkers": [ { "id": "4.21", "label": "Problem", "offset": 313 }, { "id": "4.21", "label": "Solution for Problem", "offset": 372 }, { "id": "4.22", "label": "Problem", "offset": 780 }, { "id": "4.22", "label": "Solution for Problem", "offset": 936 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0098.png", "sha256": "20b43d37cef17bc81525253fcc3e2ae0b74eb67300cc1272be0acc2aad60f861", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 99, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b53aa2640c8a5d1d45a184e04496bafdb99b6f15a3b9d5081e8378616e9c3f59", "textHash": "e3fb844cb2f02d0ac18fa66ac2a0a2e8f82c64ebd9df7ac5f7a3171e57d31ba4", "characters": 1757, "flags": [], "candidateHeadings": [ "Exercises", "4.8 Summary" ], "candidateMarkers": [ { "id": "4.23", "label": "Problem", "offset": 31 }, { "id": "4.23", "label": "Solution for Problem", "offset": 119 }, { "id": "4.7.1", "label": null, "offset": 1087 }, { "id": "4.7.2", "label": null, "offset": 1254 }, { "id": "4.7.3", "label": null, "offset": 1317 }, { "id": "4.7.4", "label": null, "offset": 1372 }, { "id": "4.8", "label": null, "offset": 1528 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0099.png", "sha256": "cc383ee3fd34cb49bf3e8f5611a1818dd86cd4cb4bc697cd44206e2ea1925c31", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 100, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "12909177c9a4a2daaa3aca95d2f2846a2769afcc5e422d800d01f616848ca949", "textHash": "d8a173d9487d9757e0723ab619f2c062fc098ba8a70bafcf381fae0a79e0dc06", "characters": 1944, "flags": [], "candidateHeadings": [ "Definitions:", "Important: The Fundamental Theorem of Arithmetic states that every positive integer has exactly one prime factorization.", "Important: Prime factorization helps us solve many problems involving integers by giving us a nice way to organize important information about each integer." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0100.png", "sha256": "1e947e16b8162243769e95c85790eb0a62f43426147ab7f0378676b0fd244fcd", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 101, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0f1ed8d3aa9fb70fc323ea62f8ad1ffb658de2e11d57b2513d06128a913fda71", "textHash": "c3de6001cef09be47fd9bf5eb172538af35c11a956a57d473987a7b791eb707f", "characters": 1785, "flags": [], "candidateHeadings": [ "Important: Let a and b be two positive integers. Let n be the number of prime factors of a or b. Let \\(p_1, p_2, \\ldots, p_n\\) be those prime factors. Now we write a and b in terms of those prime factors:", "Important: For any positive integers a, b, and c," ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0101.png", "sha256": "f9ca7ec2fe5355bb1c138969b4b9c466fdbbc10a030be907b0f10661d9dbd351", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 102, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3d402e3565b8acf893a371fc71485368f16ee143806b064862098c7fc1ba0f66", "textHash": "3da094196543f032b31a98f7d0828ee3667db895343be5f58c886c4e61af7842", "characters": 1299, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "REVIEW PROBLEMS", "4.24 Find the prime factorization of each of the following integers.", "4.25 Compute each of the following.", "4.26 Find the 8 smallest positive common multiples of 54 and 99.", "4.27 Find the smallest four-digit number that is divisible by each of the four smallest prime numbers.", "4.28 List all of the positive common divisors of 84 and 132.", "4.29 List all of the positive common divisors of 162 and 540.", "4.30 Express each of the following rational numbers either as an integer or as a fraction in lowest terms.", "4.31 Find the five smallest perfect squares that are positive multiples of 15.", "4.32 Find the six smallest perfect cubes that are positive multiples of 49.", "4.33 Find the four smallest positive multiples of 6, each of which is either a perfect square or a perfect cube.", "4.34 Find the product of the GCD and the LCM of 18 and 42.", "4.35 The GCD of n and 216 is 24 and their LCM is 864. Find n." ], "candidateMarkers": [ { "id": "4.24", "label": null, "offset": 33 }, { "id": "4.25", "label": null, "offset": 150 }, { "id": "4.26", "label": null, "offset": 363 }, { "id": "4.27", "label": null, "offset": 429 }, { "id": "4.28", "label": null, "offset": 532 }, { "id": "4.29", "label": null, "offset": 593 }, { "id": "4.30", "label": null, "offset": 655 }, { "id": "4.31", "label": null, "offset": 906 }, { "id": "4.32", "label": null, "offset": 986 }, { "id": "4.33", "label": null, "offset": 1062 }, { "id": "4.34", "label": null, "offset": 1175 }, { "id": "4.35", "label": null, "offset": 1234 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0102.png", "sha256": "b61db73f1645cd71dcef5e6c590584c1968003e87426585c6725cc05016a659d", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 103, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f148f9e2721ebb0a014d55af016cfcc20d83e3514931ca135739a5d29fbcf376", "textHash": "cb047c1b852db763d2c07afbab48dd7d6802fedf2a6fe82f2739fdda960a8d0e", "characters": 2102, "flags": [], "candidateHeadings": [ "Challenge Problems", "4.36 Calculate gcd(2520, 7200).", "4.37 Find the prime factorization of 999,999.", "4.38 The least common multiple of 12, 15, 20 and k is 420. What is the least possible value of k? (Source: MATHCOUNTS) Hints: 74", "4.41 Find the five smallest positive common multiples of 18 and 30 that are both perfect squares and perfect cubes.", "4.43 Find the greatest natural number that must be a divisor of any common multiple of 14, 26, and 66. Hints: 118", "4.44 Which of 1999, 2000, or 2001 has the largest proper divisor? (Source: Mandelbrot) Hints: 9", "4.46 The larger of two integers is 8 times the smaller. Explain why they can’t both be perfect squares.", "4.47 The sum of the reciprocals of three consecutive integers is 47/60. What is the sum of the three integers? Hints: 135", "4.48 Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her two stamp books. One of her stamp books has a total of 840 stamps. The other has 1008. What is the largest number of stamps than Jenna could be putting on each page? Hints: 48", "4.49 What is the greatest prime divisor of the following arithmetic series:" ], "candidateMarkers": [ { "id": "4.36", "label": null, "offset": 51 }, { "id": "4.37", "label": null, "offset": 84 }, { "id": "4.38", "label": null, "offset": 131 }, { "id": "4.39", "label": null, "offset": 261 }, { "id": "4.40", "label": null, "offset": 734 }, { "id": "4.41", "label": null, "offset": 849 }, { "id": "4.42", "label": null, "offset": 966 }, { "id": "4.43", "label": null, "offset": 1072 }, { "id": "4.44", "label": null, "offset": 1187 }, { "id": "4.45", "label": null, "offset": 1284 }, { "id": "4.46", "label": null, "offset": 1456 }, { "id": "4.47", "label": null, "offset": 1561 }, { "id": "4.48", "label": null, "offset": 1684 }, { "id": "4.49", "label": null, "offset": 1980 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0103.png", "sha256": "1d99471a06981ebd51ea7999f3217941fc4707131a061a8188566dbf4ad4824c", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 104, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f7a2e1980eeb5b51f30b33d75391357e5eaecefe1a5e75746afe46372e1059c4", "textHash": "a803ac6b19a8aab8be622e1ac9bb2cb9ba7d6eed9abe9a0cf282c0826872aea3", "characters": 2902, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "4.51 What is the least possible whole number that can be multiplied by 200 such that the product is a perfect cube? (Source: MATHCOUNTS)", "4.52 Jayne writes the integers from 1 to 2000 on a piece of paper. She erases all the multiples of 3, then all the multiples of 5, and so on, erasing all the multiples of each odd prime. How many numbers are left when she finishes? (Source: Mandelbrot) Hints: 142", "4.54 Alex counted to 400 by 6's beginning with 6, and Matthew counted to 400 by 4's beginning with 4. How many of the numbers counted by Alex were also counted by Matthew? (Source: MATHCOUNTS)", "4.55 The product of any two of the positive integers 30, 72, and N is divisible by the third. What is the smallest possible value of N? (Source: Mandelbrot) Hints: 83", "4.56 For two natural numbers, n and m, the GCD of n and m is 6 and the LCM of n and m is 600." ], "candidateMarkers": [ { "id": "4.50", "label": null, "offset": 19 }, { "id": "4.51", "label": null, "offset": 175 }, { "id": "4.52", "label": null, "offset": 313 }, { "id": "4.53", "label": null, "offset": 578 }, { "id": "4.54", "label": null, "offset": 726 }, { "id": "4.55", "label": null, "offset": 920 }, { "id": "4.56", "label": null, "offset": 1088 }, { "id": "4.57", "label": null, "offset": 1557 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0104.png", "sha256": "07526edd22e4cfea203bf8d6f935e11fbbda4f149a6e48f7f2f609e1d7acc320", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 105, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0ed3d526c4eaec53b789460523492cbbbad7c706cf6623ebe6a210832a62eb0a", "textHash": "c9bae82b953bf96745edc635214a5df2145da5a2bc59f76e7acbb820ebabf90a", "characters": 2326, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/page-0105.png", "sha256": "6cd688a413fccb30ae8059c1758693272609fefc641155fcded706bd13310722", "nextChapterHeadingPDF": 106 } }, { "source": "book-05", "pdfPage": 106, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "94d7595340c4b20bdc7fe8c1a7bf68e45e58e3c4dc201652f34cd7292a1d2903", "textHash": "60fc5ddf0f823d342f9c2d372bd1b3934582e1f6029495f1599affc3408095cb", "characters": 914, "flags": [], "candidateHeadings": [ "5.1 Introduction", "5.2 Counting Divisors" ], "candidateMarkers": [ { "id": "5.1", "label": null, "offset": 119 }, { "id": "5.2", "label": null, "offset": 597 }, { "id": "5.1", "label": "Problem", "offset": 630 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0106.png", "sha256": "ef6ee8ada6588676fa1347b70010c96b4a6eb29baa6d3a4274b7b0036c8d0ca5", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 107, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f8d63b7f0c66c91f99bfdd1c301f918a66bf91ea3c49e4f57e5d6a35adb022f7", "textHash": "a6a026848f20119f48d29da4c84b90c4f173228a7c48bfdf229e63f0302eed22", "characters": 6914, "flags": [ "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.2", "label": "Problem", "offset": 28 }, { "id": "5.3", "label": "Problem", "offset": 90 }, { "id": "5.1", "label": "Problem", "offset": 272 }, { "id": "5.1", "label": "Solution for Problem", "offset": 334 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0107.png", "sha256": "bd8ec7db3780a893361f6f8274800d959b7cfac6b9f243fc433e599ea4eaaf3c", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 108, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f7c5f6b00b963af84fd61066e9835715d3bfc4ef3ef9b658de3be083bbdc1084", "textHash": "087f3cf1964e9b0c9cdfaec7789014cf8f153bf1ce8c99216c994aaab269ac8c", "characters": 2253, "flags": [], "candidateHeadings": [ "Important: Since giving a quantity a name makes it easier to talk about, we denote the number of positive divisors of a natural number n as \\(t(n)\\).", "Important: Let n be a natural number with prime factorization,", "Concept: While it is convenient to know mathematical formulas (such as the one we derived for counting divisors), it is far more important to understand the method used. Over the course of this chapter we examine many divisor counting problems (where the formula we developed does not apply) that require creative solutions involving the method we developed." ], "candidateMarkers": [ { "id": "5.2", "label": "Problem", "offset": 174 }, { "id": "5.2", "label": "Solution for Problem", "offset": 236 }, { "id": "5.3", "label": "Problem", "offset": 520 }, { "id": "5.3", "label": "Solution for Problem", "offset": 697 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0108.png", "sha256": "eb8dbefe76d70e628e832c2041ad14f1d2f79a22536f514c4dc132369fa96f53", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 109, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ce1739ec85091fe7f915ffaeaf4c1059af07be38f6bddc28482e9db3933ef10a", "textHash": "de4ef3feed405f79f3bdebe739abe14dce9c14e41aef160227d34b4cc43ac2cc", "characters": 1642, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "5.2.1", "label": null, "offset": 39 }, { "id": "5.2.2", "label": null, "offset": 217 }, { "id": "5.2.3", "label": null, "offset": 270 }, { "id": "5.3", "label": null, "offset": 507 }, { "id": "5.4", "label": "Problem", "offset": 855 }, { "id": "5.5", "label": "Problem", "offset": 1091 }, { "id": "5.6", "label": "Problem", "offset": 1315 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0109.png", "sha256": "8f3848e9bb7c19e03ef43b10a4c7514a0d385fbc4d1b791c2e9062eced63a90d", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 110, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a63c7d1119b45985c1b6ac011cc024d6e9f8fb58fb6e8dff52385644757a8d4d", "textHash": "6650f0c48fab21ee487a8f95bbdc85ee2b0333e16d5e1539fe9f0b289480532a", "characters": 2326, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.3", "label": null, "offset": 0 }, { "id": "5.7", "label": "Problem", "offset": 32 }, { "id": "5.8", "label": "Problem", "offset": 207 }, { "id": "5.9", "label": "Problem", "offset": 693 }, { "id": "5.10", "label": "Problem", "offset": 910 }, { "id": "5.4", "label": "Problem", "offset": 1255 }, { "id": "5.4", "label": "Solution for Problem", "offset": 1728 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0110.png", "sha256": "a4bc4e53fddb04108a08c504c4f9e71ad2a33a6120075fecbb1824b5a0c0df0e", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 111, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fca11a54fb25f625ce0fde8c5faf8b7bdf1d115173e898a9d172cfeb49c43070", "textHash": "3cfacd27066126b95e1bac217ab28998b36693d8597a0dbeb00be16abe997ad8", "characters": 2873, "flags": [], "candidateHeadings": [ "Concept: A second solution to this problem that involves a technique we call “complementary counting.” Complementary counting is the process of counting what we do not want (instead of what we do want) and subtracting what we don’t want from the whole:" ], "candidateMarkers": [ { "id": "5.5", "label": "Problem", "offset": 1122 }, { "id": "5.5", "label": "Solution for Problem", "offset": 1214 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0111.png", "sha256": "3119b8526d384d71fab7690bdb9932cfda59318ece76a1e2bd304d59ef00d6fd", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 112, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c2d46f588605b9c0d6975d052d903e71c0ff49566e73346a2bf4fc5187671305", "textHash": "8122359d50ff566e4d0b03617e8009c888eb91ef835b0c9390d43a7f02bc1715", "characters": 2643, "flags": [], "candidateHeadings": [ "Important: A positive integer is a perfect square if and only if it has an odd number of positive divisors." ], "candidateMarkers": [ { "id": "5.3", "label": null, "offset": 0 }, { "id": "5.6", "label": "Problem", "offset": 1593 }, { "id": "5.6", "label": "Solution for Problem", "offset": 1689 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0112.png", "sha256": "bad4c988b208313c7b7040a76dea8be012a7420e8ec8138af43c6844d178825c", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 113, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f7d52e0da503f583d89a58bc3f9ea1e182f94df7e2d72260c713492c9dfebe0f", "textHash": "67beb37cb9db73fd7cb3d63855bcd8c8ed15f7b87228d9cd15e2ab109ab19b5c", "characters": 2940, "flags": [], "candidateHeadings": [ "Concept: Organizing information mathematically helps you more easily work toward a solution to a problem." ], "candidateMarkers": [ { "id": "5.7", "label": "Problem", "offset": 1058 }, { "id": "5.7", "label": "Solution for Problem", "offset": 1160 }, { "id": "5.8", "label": "Problem", "offset": 1697 }, { "id": "5.8", "label": "Solution for Problem", "offset": 1879 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0113.png", "sha256": "3655cfe5313123b2b5ac96db6abcb2f6797794062bd6642f589816fac3983362", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 114, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b4d338945a999947163c0075f2c53b48dc39ea3932415f732dd2b0cd4e3f601b", "textHash": "ad4b1a54325b6d92f417fda188cf33bf432767cdf97e7a86ed3ee41a697bfcef", "characters": 2059, "flags": [], "candidateHeadings": [ "Concept: We first used prime factorizations of integers to help us determine divisor counts. Conversely, divisor counts themselves sometimes help us determine possible prime factorizations of integers. Understanding this relationship between the prime factorizations and divisor counts of integers gives us greater flexibility in solving problems." ], "candidateMarkers": [ { "id": "5.3", "label": null, "offset": 0 }, { "id": "5.9", "label": "Problem", "offset": 784 }, { "id": "5.9", "label": "Solution for Problem", "offset": 865 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0114.png", "sha256": "b824d8dd7eeed31d7549c2b042b463cc3a3b40730addf674d34f4e5591f96414", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 115, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5de8b4159502db4f0d9ade923e0c309733348ddf7fad160e2223b707a3a2837a", "textHash": "dbc47de8cb4010e1a71fe03508278f622addb81c6a458a43e3ab6d7561fe121b", "characters": 2466, "flags": [], "candidateHeadings": [ "Concept: Check your solutions to be sure that they solve the problem. This is even more important with unusual or difficult problems. So far, no math student was born who didn't make mistakes, but good problem solvers catch many of their own errors.", "Exercises" ], "candidateMarkers": [ { "id": "5.10", "label": "Problem", "offset": 279 }, { "id": "5.10", "label": "Solution for Problem", "offset": 436 }, { "id": "5.3.1", "label": null, "offset": 1763 }, { "id": "5.3.2", "label": null, "offset": 1821 }, { "id": "5.3.3", "label": null, "offset": 1890 }, { "id": "5.3.4", "label": null, "offset": 1989 }, { "id": "5.3.5", "label": null, "offset": 2053 }, { "id": "5.3.6", "label": null, "offset": 2119 }, { "id": "5.3.7", "label": null, "offset": 2243 }, { "id": "5.3.8", "label": null, "offset": 2392 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0115.png", "sha256": "75e1682c6c2d5bcf57165870688fd8483fbed94927a2f2818705f631d54b7602", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 116, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "efd72a6bf0862e24cd8bef68bd441ae09222226a4ce7f89a62c27d4ee5a66ce0", "textHash": "96860f227502133117f37a5e00b0095d2bf239ac37a50a7bd3287da6d2af5025", "characters": 1946, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.4", "label": null, "offset": 0 }, { "id": "5.3.9", "label": null, "offset": 321 }, { "id": "5.3.10", "label": null, "offset": 444 }, { "id": "5.3.11", "label": null, "offset": 541 }, { "id": "5.4", "label": null, "offset": 677 }, { "id": "5.11", "label": "Problem", "offset": 849 }, { "id": "5.12", "label": "Problem", "offset": 1151 }, { "id": "5.11", "label": "Problem", "offset": 1318 }, { "id": "5.11", "label": "Solution for Problem", "offset": 1410 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0116.png", "sha256": "4afd5f6a78bbb805ff39bdedac0014a0a8cfcb4ed6fd85b175d55ee328d64db2", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 117, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a18894455df298aecb98b6ac401be70e9cffe3996eeb996c494c27fd46399a36", "textHash": "3162e5e991cf99d8d039ce3eb3fbe66cbf5670c902e54424aeb7a8df19493a38", "characters": 2452, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.12", "label": "Problem", "offset": 1961 }, { "id": "5.12", "label": "Solution for Problem", "offset": 2057 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0117.png", "sha256": "e7eb8de32440bb7fee89c3de09d5fa8d1f8c24c1e2014b85c5e95c4bffff0872", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 118, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2ea93cc65148da3526289995b05dccb9df55236798db51929197a8624401d521", "textHash": "5464d61451188c018245303a5f25c3d1a691c497dd07c1707e7a5e7026a42e7a", "characters": 2393, "flags": [], "candidateHeadings": [ "Concept: Identifying the differences between an unusual or difficult problem and more ordinary problems often helps us determine how we can use the methods for solving the ordinary problems to solve the more unusual and difficult ones. This highlights the importance of understanding each problem solving method well enough to use it flexibly.", "Exercises" ], "candidateMarkers": [ { "id": "5.4", "label": null, "offset": 0 }, { "id": "5.4.1", "label": null, "offset": 2276 }, { "id": "5.4.2", "label": null, "offset": 2332 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0118.png", "sha256": "a176615a9ed40693ed710b4008f02b8615151f7bb02558024de5cac5c7b306ad", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 119, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "26345e226e11f43346fd25af3aaa2769f81f4be5f347a0e3e3a178957cf70b2f", "textHash": "a9126c391288ec40244141f3ce299f84302f1118a3cf23a556f06072e992e269", "characters": 2149, "flags": [], "candidateHeadings": [ "5.5 Summary", "Concept: Prime factorization provides us with an organized approach to divisor counting problems.", "Concept: It is easier to talk about a quantity if we give it a name.", "Concept: While it is convenient to know the formulas for problems such as divisor counts, it is far more important to understand the methods involved." ], "candidateMarkers": [ { "id": "5.4.3", "label": null, "offset": 28 }, { "id": "5.4.4", "label": null, "offset": 85 }, { "id": "5.4.5", "label": null, "offset": 141 }, { "id": "5.4.6", "label": null, "offset": 287 }, { "id": "5.4.7", "label": null, "offset": 349 }, { "id": "5.4.8", "label": null, "offset": 431 }, { "id": "5.5", "label": null, "offset": 505 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0119.png", "sha256": "daf8f690114b63e3fca6f2701d7392c28271150df30021e50c12cbb6d1d8a6c3", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 120, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8b6c1bdf3378d2fb4b8696198710186fac8353f9bc6d79eb7a87a46bdf511728", "textHash": "6fe4987f65e70373c5accf61ef0d70a13bef732f6f7bfb085f45948c87b66022", "characters": 2058, "flags": [], "candidateHeadings": [ "Important: A positive integer is a perfect square if and only if it has an odd number of positive divisors.", "Concept: It is sometimes easier to count the things we don't want than the things we do want. Complementary counting is the process of counting what we do not want (instead of what we do want) and subtracting what we don't want from the whole.", "Concept: Organizing information mathematically helps you more easily work toward a solution to a problem.", "Concept: We first used prime factorizations of integers to help us determine divisor counts. Conversely, divisor counts themselves sometimes help us determine possible prime factorizations of integers. Understanding this relationship between the prime factorizations and divisor counts of integers gives us greater flexibility in solving problems.", "Concept: Check your solutions—particularly those to unusual or difficult problems." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0120.png", "sha256": "f5143b0d4daf71be0192d517b235e23919a2852b46617818b86e80c76cc4f87a", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 121, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "dc339ea7a07e95fac7eada6d6372d482a5464d221f5c74e0bf276d4099644ddb", "textHash": "8aec515847b87b15699dff1c031a1857331e51e5abc533f9874e11658f9b7f0f", "characters": 1861, "flags": [], "candidateHeadings": [ "Concept: Identifying the differences between an unusual or difficult problem and more ordinary problems often helps us determine how we can use the methods for solving the ordinary problems to solve the more unusual and difficult problems.", "REVIEW PROBLEMS", "5.13 Find the number of positive divisors of each of the following.", "5.14 Find the number of distinct positive divisors of (30)⁴ excluding 1 and (30)⁴. (Source: AHSME)", "5.15 Joseph and Timothy play a game in which Joseph picks an integer between 1 and 1000 inclusive and Timothy divides 1000 by that integer and states whether or not the quotient is an integer. How many integers could Joseph pick such that Timothy’s quotient is an integer?", "5.16 How many of the positive divisors of 3240 are multiples of 3?", "5.17 How many of the divisors of 3240 are perfect squares?", "5.18 Find the product of the positive divisors of each of the following.", "5.19 Find the product of the even positive divisors of 240." ], "candidateMarkers": [ { "id": "5.13", "label": null, "offset": 1072 }, { "id": "5.14", "label": null, "offset": 1189 }, { "id": "5.15", "label": null, "offset": 1289 }, { "id": "5.16", "label": null, "offset": 1563 }, { "id": "5.17", "label": null, "offset": 1631 }, { "id": "5.18", "label": null, "offset": 1691 }, { "id": "5.19", "label": null, "offset": 1796 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0121.png", "sha256": "5edffa577d24ed1702eb7ec0705d3f184ed932bb1e68434acca8301185b7da3c", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 122, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "16d4a50e47598f7c39dbe33dda78271ee7f5554e949bb2921dafe71292f86c17", "textHash": "5afae96b8dcf64dd07184b30e089064ea044454a760df74a31e1c8eeb1831751", "characters": 2204, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "Challenge Problems", "5.20 How many positive integers are divisors of 999,999?", "5.21 Let m and n be two relatively prime natural numbers. If m has 12 positive divisors and n has 10 positive divisors, what is the product of the positive divisors of mn? Hints: 42", "5.22 The total number of positive divisors of an integer is prime. How many prime numbers are divisors of the integer? Hints: 97", "5.23 What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors? (Source: MATHCOUNTS) Hints: 47", "5.25 If n has 2 prime divisors and 9 total divisors, how many divisors does n² have? Hints: 14", "5.26 Find the smallest positive integer that can be written as a product of two factors (each greater than 1) in exactly three different ways. For example, 70 is one such integer since \\( 70 = 2 \\cdot 35 = 5 \\cdot 14 = 7 \\cdot 10 \\). Note that the order of the factors does not matter. (Source: Mandelbrot) Hints: 110", "5.27 How many of the positive divisors of 14400 themselves have exactly 8 positive divisors?", "5.28 How many positive divisors do 8400 and 7560 have in common?", "5.29 A certain integer has 20 positive divisors.", "5.30 For some natural number n, the integer 6n has 9 positive divisors.", "5.31 What is the sum of all positive integers less than 100 that have exactly twelve divisors? (Source: Mandelbrot) Hints: 150" ], "candidateMarkers": [ { "id": "5.20", "label": null, "offset": 39 }, { "id": "5.21", "label": null, "offset": 97 }, { "id": "5.22", "label": null, "offset": 280 }, { "id": "5.23", "label": null, "offset": 410 }, { "id": "5.24", "label": null, "offset": 549 }, { "id": "5.25", "label": null, "offset": 874 }, { "id": "5.26", "label": null, "offset": 970 }, { "id": "5.27", "label": null, "offset": 1289 }, { "id": "5.28", "label": null, "offset": 1383 }, { "id": "5.29", "label": null, "offset": 1449 }, { "id": "5.30", "label": null, "offset": 1852 }, { "id": "5.31", "label": null, "offset": 2072 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0122.png", "sha256": "b00ad8487c4fe2073baaf67bec79740c02a3a58676d6f493d3cd3a962e4975a3", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 123, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b7d3005f47ef30ae8ea05946b3575990211ef86c8b00e37cb42acb80c239e226", "textHash": "c791d660a907049c56070558127e7209109b145a1b5fbccb2ffd0ed0e35d78da", "characters": 1401, "flags": [], "candidateHeadings": [ "5.34 Find the product of the positive divisors of 6480 that are multiples of 12. Hints: 165" ], "candidateMarkers": [ { "id": "5.32", "label": null, "offset": 28 }, { "id": "5.33", "label": null, "offset": 830 }, { "id": "5.34", "label": null, "offset": 955 }, { "id": "5.35", "label": null, "offset": 1048 }, { "id": "5.36", "label": null, "offset": 1270 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/page-0123.png", "sha256": "2c1d8ab48aa82324839e04f04e6e4b37ba8584409a868765827c93a76522ceec", "nextChapterHeadingPDF": 124 } }, { "source": "book-05", "pdfPage": 124, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a7250f6d3b20646b44dcd15ec823ae97d52f26bdfa63c89afe8ebb214b1b2bbd", "textHash": "ec5268f3e7ad9d14eb8f0636f412052372679014865a60c1b90e8cab46fe662f", "characters": 1015, "flags": [], "candidateHeadings": [ "6.1 Introduction", "6.2 Some Special Primes" ], "candidateMarkers": [ { "id": "6.1", "label": null, "offset": 94 }, { "id": "6.2", "label": null, "offset": 419 }, { "id": "6.1", "label": "Problem", "offset": 613 }, { "id": "6.2", "label": "Problem", "offset": 674 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0124.png", "sha256": "c1afe7eba0a716a7c010110354c39aa28574261ec763bc3d64f27633db05a111", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 125, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c9049cd10dffb828738df262de02ee61882e29d2b7d919768022c4ca3b1781d3", "textHash": "eb031ff6dde7ac18d150723e4e5a4c114cc0295e68688fb132387c71eac750b7", "characters": 2650, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "6.1", "label": "Problem", "offset": 1191 }, { "id": "6.1", "label": "Solution for Problem", "offset": 1262 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0125.png", "sha256": "c43f5328060a7eba29eda28c10582a422f6cf666bbb3d9254fae2595175b7159", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 126, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "30e946f785e004013fc4f0f663e65edce8aafd296ac4ecdde5c236de38e82c9a", "textHash": "c47f8aed4e1bc309aa89ca83de9e6fdc32e308ba659264e52b912f06bc32e8f0", "characters": 2169, "flags": [], "candidateHeadings": [ "Definition: A conjecture is a statement that is believed to be true, but has not been proven to be true. When a conjecture is proven, it is then called a theorem.", "Exercises", "6.3 Factorials, Exponents and Divisibility", "Definition: The factorial of a natural number \\(n\\) is the product of all the natural numbers less than or equal to \\(n\\). We write this as \\(n!\\):" ], "candidateMarkers": [ { "id": "6.2", "label": "Problem", "offset": 1168 }, { "id": "6.2", "label": "Solution for Problem", "offset": 1238 }, { "id": "6.2.1", "label": null, "offset": 1511 }, { "id": "6.2.2", "label": null, "offset": 1543 }, { "id": "6.3", "label": null, "offset": 1611 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0126.png", "sha256": "755a66ee785fce92cb07e94882ce49a59a067f4748b5577dbea2d68c6fc68a6a", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 127, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "428bd9707445b57bc332166d2aa06ddab4c5165d0e0d25728a5c5eb480b19dd9", "textHash": "575ed3b99d951c1a1d5e53c302666a6bb896d90ac1f01737b8bad1c3768c51db", "characters": 1298, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "6.3", "label": "Problem", "offset": 162 }, { "id": "6.4", "label": "Problem", "offset": 189 }, { "id": "6.5", "label": "Problem", "offset": 318 }, { "id": "6.3", "label": "Problem", "offset": 893 }, { "id": "6.3", "label": "Solution for Problem", "offset": 920 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0127.png", "sha256": "d9d26e927f3d7225379a232534bfe0b649de08a2f50c526046a3eab14a15a8b0", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 128, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c7e13752a8d02bf32ca7bebece32998b21984f422121353dd0b0cad580acb45d", "textHash": "8e74fa5db0cce9bba74850122bc8131d776c46a15d81df7a540353632d59dcdb", "characters": 2295, "flags": [], "candidateHeadings": [ "Important: Because of the way factorials are defined, there is a simple relationship between consecutive factorials:", "Concept: Many arithmetic and number theory problems are simplified using algebraic tactics. In Problem 6.4 we used factorization to turn the sum 5! + 6! into the product 7·5!." ], "candidateMarkers": [ { "id": "6.4", "label": "Problem", "offset": 374 }, { "id": "6.4", "label": "Solution for Problem", "offset": 431 }, { "id": "6.5", "label": "Problem", "offset": 1108 }, { "id": "6.5", "label": "Solution for Problem", "offset": 1168 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0128.png", "sha256": "ec2e51df95d4f1fcfb651dbb34086ccc183f3ce15203c629e328eb9ec3e0df0b", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 129, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "80f2f41b0d3926d131376986762c78dc0a3ec04a7f0be0d1864dc1ccf0cee1c7", "textHash": "f4f1d447d28f8a40881d46e91314d9506a6feddbb8684d06522a31995f8ba182", "characters": 6105, "flags": [ "truncated" ], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0129.png", "sha256": "01bedf2042264d88d931af12426f67ba8773eee5671c4f72e61417b541b86b8a", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 130, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "237b4f8ec19722ef1e4b6d489741e6f30241dc3c40b3162c9833cfaf03036c17", "textHash": "bb9100d1348b327c33d1b8009f7218484f135821c57f811d4aa0bad57f4e86ad", "characters": 2022, "flags": [], "candidateHeadings": [ "Exercises", "6.4 Perfect, Abundant, and Deficient Numbers" ], "candidateMarkers": [ { "id": "6.3.1", "label": null, "offset": 1200 }, { "id": "6.3.2", "label": null, "offset": 1261 }, { "id": "6.3.3", "label": null, "offset": 1305 }, { "id": "6.3.4", "label": null, "offset": 1384 }, { "id": "6.3.5", "label": null, "offset": 1436 }, { "id": "6.3.6", "label": null, "offset": 1478 }, { "id": "6.3.7", "label": null, "offset": 1542 }, { "id": "6.3.8", "label": null, "offset": 1605 }, { "id": "6.4", "label": null, "offset": 1664 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0130.png", "sha256": "5a42717d197f3921a218d49bd07cb8a3e147eaa9986da8c00386ae2d40b346ea", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 131, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fef43bbfa80da26603290dc51626670dec1959389f7c864844d185a897d4c363", "textHash": "a4d214d52e2a798bad2f82471dcaa70b0a0e846fc68c71a897ad42ed3c3a42f7", "characters": 1397, "flags": [], "candidateHeadings": [ "Definition: A perfect number is a positive integer that is equal to the sum of its proper divisors.", "Definition: An abundant number is a positive integer that is smaller than the sum of its proper divisors.", "Definition: A deficient number is an integer that is greater than the sum of its proper divisors." ], "candidateMarkers": [ { "id": "6.6", "label": "Problem", "offset": 161 }, { "id": "6.7", "label": "Problem", "offset": 231 }, { "id": "6.8", "label": "Problem", "offset": 300 }, { "id": "6.6", "label": "Problem", "offset": 1063 }, { "id": "6.6", "label": "Solution for Problem", "offset": 1115 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0131.png", "sha256": "14ccbd099c22bdb7115c67bd7385315d30ce009131eaa1b777461a8a5fb6ce6e", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 132, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2e2ac103f1de889132b84a9b581fbcaf6eeaccee6212eb26d77b101bb04a3793", "textHash": "65d476cd61a3c5ae96760bd69ef5c776ed4d656491fa07f594c42131470322ad", "characters": 1399, "flags": [], "candidateHeadings": [ "Exercises", "6.5 Palindromes" ], "candidateMarkers": [ { "id": "6.7", "label": "Problem", "offset": 479 }, { "id": "6.7", "label": "Solution for Problem", "offset": 532 }, { "id": "6.8", "label": "Problem", "offset": 639 }, { "id": "6.8", "label": "Solution for Problem", "offset": 692 }, { "id": "6.4.1", "label": null, "offset": 803 }, { "id": "6.4.2", "label": null, "offset": 910 }, { "id": "6.4.3", "label": null, "offset": 1024 }, { "id": "6.4.4", "label": null, "offset": 1083 }, { "id": "6.4.5", "label": null, "offset": 1131 }, { "id": "6.5", "label": null, "offset": 1215 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0132.png", "sha256": "921d2b7ff9a7b20e578f8ab2d15eba33f37f1fa5e1b2fcc09315a5617f4878a1", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 133, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b145ea42925f48384c5896e2baeafebb706ea83ff12e95bf8e495d19bc1297c9", "textHash": "e9d958b3cc388f5d8bdcefa67e1b9d2388a2a82af06bd993459a71e04f697e4a", "characters": 1256, "flags": [], "candidateHeadings": [ "Definition: A palindrome is an integer that reads the same forward as it does backward." ], "candidateMarkers": [ { "id": "6.9", "label": "Problem", "offset": 190 }, { "id": "6.10", "label": "Problem", "offset": 296 }, { "id": "6.11", "label": "Problem", "offset": 515 }, { "id": "6.9", "label": "Problem", "offset": 681 }, { "id": "6.9", "label": "Solution for Problem", "offset": 787 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0133.png", "sha256": "5d5946693ee32204c390f63ed1766a06e2c72d8d5c2bf4576259647ba5419a0e", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 134, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c3f4a5d29f512b8096ae320ac06adb7bf3f791623a9f1718982d0b75ad0d49a7", "textHash": "3eb2c534e17faa32c6d0bd0d740cefac24dc4967b1f2d6bd71a43ac81b6dd4c5", "characters": 2244, "flags": [], "candidateHeadings": [ "Concept: Identifying the interesting and useful features of a problem often helps us find the simplest solution." ], "candidateMarkers": [ { "id": "6.10", "label": "Problem", "offset": 491 }, { "id": "6.10", "label": "Solution for Problem", "offset": 550 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0134.png", "sha256": "8188472b167b400fae6cc57b81db42b2fc51f89f62950458d18a938e3111aaaf", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 135, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b626a9111bdf6cb7bf1130c292fc5e218addbc44626de248285147986aed4c90", "textHash": "24919ee22a57e8e8a2380fa6272dae1fa4bfdfb027ce31fc9de2fcea9ccec09e", "characters": 2066, "flags": [], "candidateHeadings": [ "Exercises", "6.6 Summary" ], "candidateMarkers": [ { "id": "6.11", "label": "Problem", "offset": 27 }, { "id": "6.11", "label": "Solution for Problem", "offset": 145 }, { "id": "6.5.1", "label": null, "offset": 1346 }, { "id": "6.5.2", "label": null, "offset": 1402 }, { "id": "6.5.3", "label": null, "offset": 1463 }, { "id": "6.5.4", "label": null, "offset": 1557 }, { "id": "6.5.5", "label": null, "offset": 1621 }, { "id": "6.6", "label": null, "offset": 1725 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0135.png", "sha256": "ed39b1d3f128db2460d2b9eb4da8b0e5973817811d8d6a7eb046ce6d53fd3379", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 136, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "336b0285fa108a8ed0c4c354839cf2d5f2f3e86a7880e926d1dda05b4a5b7e4c", "textHash": "0f0eddc21c56b3c29bdd6e0f9a3a350ee6abed62f7acbefa197a9af4bdd707df", "characters": 1982, "flags": [], "candidateHeadings": [ "Definitions:", "Important: Because of the way factorials are defined, there is a simple relationship between consecutive factorials:", "Concept: Many arithmetic and number theory problems are most efficiently solved using algebraic tactics." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0136.png", "sha256": "4403a339b06f0a67fb56db3d4ce075780048baba81808973b1491174923aad73", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 137, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ddf6ea31445f870b79eb92678f820befbe8567cc51d037fbc39348cb6067503a", "textHash": "39128845145f5ebb37c3bd51268e6946d3063f7134bce2611b5caa7cb4078db4", "characters": 1165, "flags": [], "candidateHeadings": [ "Concept: Identifying the interesting and useful features of a problem often helps us find the simplest solution.", "REVIEW PROBLEMS", "6.12 Find a pair of twin primes that are each greater than 200 but less than 250.", "6.13 How many positive divisors of 30! are prime?", "6.14 For what value of n does 6! = 2^4 \\cdot 3^2 \\cdot n? (Source: MATHCOUNTS)", "6.16 Find the GCD of 7! and (5!)^2.", "6.17 Find the prime factorization of 12!.", "6.18 Which of the following natural numbers are abundant?", "6.19 Find the largest power of 3 that divides 60!.", "6.20 Identify all of the abundant numbers between 70 and 80 inclusive.", "6.21 Find the number of even three-digit palindromes.", "6.22 Find the number of five-digit palindromes.", "6.23 Find the second smallest palindrome that is an abundant number." ], "candidateMarkers": [ { "id": "6.12", "label": null, "offset": 271 }, { "id": "6.13", "label": null, "offset": 354 }, { "id": "6.14", "label": null, "offset": 405 }, { "id": "6.15", "label": null, "offset": 485 }, { "id": "6.16", "label": null, "offset": 694 }, { "id": "6.17", "label": null, "offset": 731 }, { "id": "6.18", "label": null, "offset": 774 }, { "id": "6.19", "label": null, "offset": 863 }, { "id": "6.20", "label": null, "offset": 915 }, { "id": "6.21", "label": null, "offset": 987 }, { "id": "6.22", "label": null, "offset": 1042 }, { "id": "6.23", "label": null, "offset": 1091 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0137.png", "sha256": "fc1a54a533b5aecc5c7e0adefc8f768d4c6ca63611b1e56981335985090009f3", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 138, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1b4ffaaf61ec04ce978c8b59cc110f36bf0bfb0d20a7c7692305d3ebfb6bbeb5", "textHash": "f118435a87cfa92279b069100aa88d93382b4a0208518012f062e6139a6152d8", "characters": 2219, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "Challenge Problems", "6.24 Find the smallest positive integer that is both a perfect power of 11 and is not a palindrome.", "6.25 Find the largest integer \\(n\\) such that \\(n\\) has exactly 4 positive divisors and \\(n\\) divides 100! Hints: 62", "6.26 In how many zeros does the decimal expansion of \\(100^{100} - 100!\\) end?", "6.27 How many positive cubes divide 3! - 5! - 7!? (Source: AMC)", "6.28 Explain why every prime number is deficient.", "6.29 If we set \\(C = 19!\\), then express 21! - 20! in terms of \\(C\\). (Source: Mandelbrot) Hints: 32", "6.30 Which is greater, the total number of five-digit palindromes or the total number of six-digit palindromes? Hints: 170", "6.32 Explain why 3 and 5 are the only pair of twin primes whose sum is not a multiple of 3.", "6.34 Find the total number of ten-digit palindromes. Hints: 128", "6.35 What is the largest integer value of \\(n\\) for which \\(8^n\\) evenly divides 100!? (Source: MATHCOUNTS)" ], "candidateMarkers": [ { "id": "6.24", "label": null, "offset": 39 }, { "id": "6.25", "label": null, "offset": 140 }, { "id": "6.26", "label": null, "offset": 257 }, { "id": "6.27", "label": null, "offset": 336 }, { "id": "6.28", "label": null, "offset": 400 }, { "id": "6.29", "label": null, "offset": 450 }, { "id": "6.30", "label": null, "offset": 551 }, { "id": "6.31", "label": null, "offset": 674 }, { "id": "6.32", "label": null, "offset": 810 }, { "id": "6.33", "label": null, "offset": 902 }, { "id": "6.34", "label": null, "offset": 1020 }, { "id": "6.35", "label": null, "offset": 1084 }, { "id": "6.36", "label": null, "offset": 1202 }, { "id": "6.37", "label": null, "offset": 1728 }, { "id": "6.38", "label": null, "offset": 1874 }, { "id": "6.39", "label": null, "offset": 2047 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0138.png", "sha256": "db049f74168d4b62e0e18ca40b3fa77f2dfe2af99150770941b789ca2e50e678", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 139, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d72b42e9de1a94f1f198cec7602f5d8825c2eb2fa2f23ef80a571816cb0ae2ac", "textHash": "2c2a73f8b0c1a50d983619096f673b19acced89666660daf84f7b27a76731bef", "characters": 2051, "flags": [], "candidateHeadings": [ "6.40 Let a and b be positive integers. If \\(a!/b!\\) is a multiple of 4 but not a multiple of 8, then what is the largest possible value for \\(a - b\\)? (Source: Mandelbrot) Hints: 57" ], "candidateMarkers": [ { "id": "6.40", "label": null, "offset": 27 }, { "id": "6.41", "label": null, "offset": 210 }, { "id": "6.42", "label": null, "offset": 356 }, { "id": "6.43", "label": null, "offset": 464 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/page-0139.png", "sha256": "4766537f67c69fa4209bf4473a6247d1c8ca94ee67cb4f82b857aedb53290ab2", "nextChapterHeadingPDF": 140 } }, { "source": "book-05", "pdfPage": 140, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7707a4e516a1f4c418aadebca9b1e088580ceb4f9ef5353ce0fab51153c18af2", "textHash": "1dbe5add3396f6bf2ffd71458fb8cc902fbb195091728f569893bdfedd756d0d", "characters": 1215, "flags": [], "candidateHeadings": [ "7.1 Introduction", "7.2 Problems" ], "candidateMarkers": [ { "id": "7.1", "label": null, "offset": 122 }, { "id": "7.2", "label": null, "offset": 446 }, { "id": "7.1", "label": "Problem", "offset": 470 }, { "id": "7.2", "label": "Problem", "offset": 1003 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0140.png", "sha256": "1bdd508fd7c31d58e78f9498332aa34a84844da8265a33f11dda5a63bb12d136", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 141, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d9e3f001b330ed1a8b66a392a3f59f13bdb853b241473afb2dc0b05008041bb2", "textHash": "2936d1edc2f671967824965f33685dad1f7d99ea5ecebf93e541a1a99d4fd743", "characters": 2035, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "7.3", "label": "Problem", "offset": 33 }, { "id": "7.4", "label": "Problem", "offset": 318 }, { "id": "7.5", "label": "Problem", "offset": 547 }, { "id": "7.6", "label": "Problem", "offset": 770 }, { "id": "7.7", "label": "Problem", "offset": 1058 }, { "id": "7.8", "label": "Problem", "offset": 1763 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0141.png", "sha256": "7610396b9a1e861edd4919221aa67f31555a4a11bf271cd5a44bae2e73636cb7", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 142, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "657766063ec249d0cf61b80192a57ff0517fe1f7737f506ef388048ef463de6e", "textHash": "affab4d662a853e30de14db00903943045273ed6f287d5e86f1d44225c452f2e", "characters": 2398, "flags": [], "candidateHeadings": [ "Concept: Assign variables to unknown quantities in number theory problems. This helps you apply your skill with algebra to number theory problems." ], "candidateMarkers": [ { "id": "7.9", "label": "Problem", "offset": 14 }, { "id": "7.1", "label": "Problem", "offset": 527 }, { "id": "7.1", "label": "Solution for Problem", "offset": 622 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0142.png", "sha256": "4a4e35e45234c7ff62e88f008aa86f12d84aaa83511d77569283aca98d49fec8", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 143, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "bb5e3051f82fe4608f689598c39f0043ed66c3d03969f9ea5f21abe7bfc5530b", "textHash": "c73d4603c53a5e30fe3769d1c8724f940e776ec93f646cccc03bdb0cd8bcff3d", "characters": 2159, "flags": [], "candidateHeadings": [ "Concept: When solving complex equations for integer solutions, we sometimes factor both integers and algebraic expressions to provide ourselves with similar equations to work with.", "Important:" ], "candidateMarkers": [ { "id": "7.2", "label": "Problem", "offset": 33 }, { "id": "7.2", "label": "Solution for Problem", "offset": 124 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0143.png", "sha256": "3cb45dab01bbdbf0d8eccf61955be4ac55a5ee64607fcf4b2d50caae89dedf41", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 144, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "62c892d9663d288a7cc9778796b1c2349485de300ffc027a2685b6bcdfc37162", "textHash": "567c96e9b7c6e3183d900ed0384ef21891146009f82d019f1dfacb852b2e19be", "characters": 2362, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "7.3", "label": "Problem", "offset": 14 }, { "id": "7.3", "label": "Solution for Problem", "offset": 179 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0144.png", "sha256": "1780b50775c2c3480db4c8a6ff643f9ad921bc9b79c04fac1690b3d4acc8de92", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 145, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "01f898c8f3d7b0ae5fae569c0688c3303a4a068fd71da8468fa80297c4515fd0", "textHash": "f81c83873197a163ac7addf3d67b840293ee9533d6ac9e3172065c53e245d1ab", "characters": 2420, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0145.png", "sha256": "25fdda91ffa215b50bdc50f17c7ce81dc01bb6e3b5df82277b6e356490a67481", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 146, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a71cbc6fbca27761fe8a3b8f6227db97edc915ef8708ae2905ab86671b86d830", "textHash": "e703a5f67844bbe5f56ab8c790408d420ad629fbaea52b88eaca3018b1083d9b", "characters": 2220, "flags": [], "candidateHeadings": [ "Concept: When faced with an ugly equation, try the following:", "Concept: Algebraic factorization and prime factorization are similar processes that involve writing quantities as products of simpler quantities." ], "candidateMarkers": [ { "id": "7.4", "label": "Problem", "offset": 194 }, { "id": "7.4", "label": "Solution for Problem", "offset": 365 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0146.png", "sha256": "52a01241f929b6dd78ff6c289c29a565163a7671ac4cda9218b3e6e1b942eb47", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 147, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "489a002433439f311a971c3d2276c7bd8b5ef8bfc018c11c3644d4f2d4316aee", "textHash": "3dd9ffc352bff30669d52afa21d7fc26750d68213c0d00989dcd92818c254963", "characters": 2514, "flags": [], "candidateHeadings": [ "Concept: Solving problems by any method can help you discover better methods and broaden your understanding of mathematics. Keep solving!" ], "candidateMarkers": [ { "id": "7.5", "label": "Problem", "offset": 33 }, { "id": "7.5", "label": "Solution for Problem", "offset": 136 }, { "id": "7.6", "label": "Problem", "offset": 1638 }, { "id": "7.6", "label": "Solution for Problem", "offset": 1708 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0147.png", "sha256": "72173c7748955cb184e5e8dc6d57831a34e90c23f1a6ec92f01410e2b4993836", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 148, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0f639b14937472e4421f7785521c44b62b1cd02d6764eebf3d8fffeb38cb15cf", "textHash": "49c3c15663da1f56aac732a722e012c5d395c6ffc52c000360c5970468306477", "characters": 2056, "flags": [], "candidateHeadings": [ "Definition: A generalized form of a group of numbers is an algebraic expression common to all of the numbers in the group. Variables used in generalized mathematical expressions are called parameters.", "Concept: Generalized forms of groups of numbers can help us solve problems relating to those groups." ], "candidateMarkers": [ { "id": "7.7", "label": "Problem", "offset": 1512 }, { "id": "7.7", "label": "Solution for Problem", "offset": 1768 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0148.png", "sha256": "c27c3c872a39cf865b5fff0310299d17bcdfaa8b81f1d82ffda7e5a309ad4a2a", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 149, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5b134fc24f03d85bc9570428b2405a79e8289de90a7fbe0b1a41d57d3ba7701d", "textHash": "873268ac9e6fbfb1181d5fdabffcdff77848125d84a8f3958c8ce75b0626903d", "characters": 2670, "flags": [], "candidateHeadings": [ "Concept: Carefully chosen parameters can help us simplify the algebra behind many number theory problems. The idea is that it's usually easier to solve a problem using fewer variables." ], "candidateMarkers": [ { "id": "7.8", "label": "Problem", "offset": 2147 }, { "id": "7.8", "label": "Solution for Problem", "offset": 2297 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0149.png", "sha256": "6776399845428401aba1bb823ca3e4efd62c6abf5489dcfb37534a6b01847815", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 150, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3faafa0d68469b4c62de69f0becbd0fe952209c339fd3baf9dab67d211eff461", "textHash": "7f12756f737b807c82b530a0092de0d08f3a33eb8925d5aebe15bf34195283ac", "characters": 2414, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "7.9", "label": "Problem", "offset": 1731 }, { "id": "7.9", "label": "Solution for Problem", "offset": 1827 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0150.png", "sha256": "2f9af359eede97adee88ee78d52cc1bf1ae89bdbac41c0beede12ad10e8bb9ef", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 151, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c7055ea2452107b1e9230088a2cd5277d3e907b6ccb4b820f9f6166d9ca27d79", "textHash": "3d50f14c838ff54d4bc762204a12c68516bea8b550a1541a51399d43b4dc8a17", "characters": 2242, "flags": [], "candidateHeadings": [ "Concept: Think about all the ways in which common equations can be used. A single equation that relates several variables can be used to determine the value of any one of the variables when we know the values of all the others. For instance, the equations \\(A = \\frac{8}{n}, S = An, and n = \\frac{8}{A}\\) relate the same information (so long as neither n nor A are equal to 0). However, each one isolates a different variable to make computation of the value of that variable clear." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0151.png", "sha256": "64e53852ac5e5d31d68d008663f7d031ec70a26c4b5ecb3d02810e375872d986", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 152, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e488a5d5bde00e5967c0e1020a7db69d12fbd51cd33c758251468357d99fad62", "textHash": "d604a54c6e077f9475cf9f2957c01ce21de2d250a6cc157612f1a2cd9e93c00c", "characters": 2468, "flags": [], "candidateHeadings": [ "7.3 Summary", "Definition: A generalized form of a group of numbers is an algebraic expression common to all of the numbers in the group. The variables that dictate the values of generalized expressions are parameters.", "Concept: Assign variables to unknown quantities in number theory problems. This helps you apply your skill with algebra to number theory problems.", "Concept: Carefully chosen parameters can help us simplify the algebra behind many number theory problems.", "Concept: Algebraic factorization and prime factorization are similar processes that involve writing quantities as products of simpler quantities.", "Concept: Solving problems by any method can help you discover better methods while broadening your understanding of mathematics. Keep solving!", "Important:" ], "candidateMarkers": [ { "id": "7.3", "label": null, "offset": 0 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0152.png", "sha256": "a52a166df9c83aa4ac5d09ba76ddf19a9fc88183c8a28f82f985d429be5ff3e7", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 153, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9b55861da73be4d11a8f77976475ce065441832ed76f6c5a5db781616d1c8528", "textHash": "28a459b329312e0947dd20f7e68841fcd8a1b359cacb960f6436576f83576988", "characters": 2408, "flags": [], "candidateHeadings": [ "Concept: Generalized forms of groups of numbers can help us solve problems relating to those groups.", "Challenge Problems", "7.10 How many perfect squares less than 500 are divisible by 8? (Source: MATHCOUNTS)", "7.11 Find each two-digit number that is equal to four times the sum of its digits.", "7.12 For how many positive \\(n\\) is \\(n^2 - 3n + 2\\) a prime number? (Source: AMC)", "7.13 There are three unique two-digit numbers such that the sum of the squares of the first number’s digits equals the sum of the squares of the second number’s digits equals the sum of the squares of the third number’s digits, and each of these sums is equal to the sum of the squares of the digits of 345. What is the sum of these three two-digit numbers? (Source: MATHCOUNTS)", "7.14 How many four-digit integers are multiples of 15, 20, and 25?", "7.15 John has a jar of between 500 and 600 coins that are all nickels, dimes, and quarters. The ratio of the number of quarters to the number of dimes is 8 : 5 and the ratio of the number of dimes to the number of nickels is 6 : 11. What is the total value of the money in John’s jar of coins?", "7.16 The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this property? (Source: AMC)", "7.17 The measure of each interior angle of a regular \\(n\\)-sided polygon is given by the formula \\(180^\\circ - \\frac{360^\\circ}{n}\\). For instance," ], "candidateMarkers": [ { "id": "7.10", "label": null, "offset": 519 }, { "id": "7.11", "label": null, "offset": 605 }, { "id": "7.12", "label": null, "offset": 689 }, { "id": "7.13", "label": null, "offset": 773 }, { "id": "7.14", "label": null, "offset": 1153 }, { "id": "7.15", "label": null, "offset": 1221 }, { "id": "7.16", "label": null, "offset": 1516 }, { "id": "7.17", "label": null, "offset": 1694 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0153.png", "sha256": "b82c1c796ca5178a779049e81f695a48bae23909ec6efe09bde8fc2988c3d60f", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 154, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0b3e2e3aeaad67920bdda17fcef386236531ae2ab107adf37d6a63a338b6b7f7", "textHash": "2e8bc603b9bb0709b931b2ca3b6afd283a796df62b6e3e24816d6d68e728d394", "characters": 2306, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "7.18 Define a Benson rectangular number as a positive integer \\(a\\) that, if increased and decreased by the same positive integer \\(b\\), yields two positive integers, \\(a + b\\) and \\(a - b\\), that are both perfect squares. What is the least Benson rectangular number? (Source: MATHCOUNTS)", "7.19 Find all ordered pairs of positive integers \\((x, y)\\) where", "7.20 There are 9 six-digit natural numbers that have the property that all of their digits are the same (such as 111111 and 222222). Find the largest prime number that is a divisor of all of them.", "7.21 Show that one more than the product of any pair of twin primes is a perfect square. Hints: 132", "7.22 When a positive integer \\(x\\) is divided by a positive integer \\(y\\), the quotient is \\(u\\) and the remainder is \\(v\\), where \\(u\\) and \\(v\\) are integers. What is the remainder when \\(x + 2uy\\) is divided by \\(y\\)? (Source: AMC)", "7.23 How many three-digit perfect squares are multiples of both 3 and 4?", "7.24 Two farmers agree that pigs are worth $300 and that goats are worth $210. When one farmer owes the other money, he pays the debt in pigs or goats, with “change” received in the form of goats or pigs as necessary. (For example, a $390 debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? (Source: AMC) Hints: 94", "7.26 How many four-digit integers are even perfect squares and also perfect cubes? Hints: 167", "7.27 Let \\(A, M\\), and \\(C\\) be digits with", "7.28 The difference between the squares of two natural numbers is prime. Show that the sum of the two natural numbers is also prime. Hints: 136" ], "candidateMarkers": [ { "id": "7.18", "label": null, "offset": 19 }, { "id": "7.19", "label": null, "offset": 309 }, { "id": "7.20", "label": null, "offset": 434 }, { "id": "7.21", "label": null, "offset": 632 }, { "id": "7.22", "label": null, "offset": 733 }, { "id": "7.23", "label": null, "offset": 969 }, { "id": "7.24", "label": null, "offset": 1043 }, { "id": "7.25", "label": null, "offset": 1459 }, { "id": "7.26", "label": null, "offset": 1776 }, { "id": "7.27", "label": null, "offset": 1871 }, { "id": "7.28", "label": null, "offset": 1998 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0154.png", "sha256": "dcfe72a81a2b8d04278cf59d53dacee20f87e0d34fa39e6dfa70c5f81ec4890b", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 155, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0d5bab120cce448b7ae0a0b6b92d49554a049f2963f4fb3367ac9283729c4d53", "textHash": "0a5675d99490729a8c20e6f743eb5c0af8e1e2018e3de4579a9fde7ac5ff4dd2", "characters": 1188, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/page-0155.png", "sha256": "00dfc87e6c37ea1a3cdbb8e4716dabd7c02d6e5268c4f2298db7588e56a4be5f", "nextChapterHeadingPDF": 156 } }, { "source": "book-05", "pdfPage": 156, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "340494001d87487aa7f476cd42d32b52680243073e446ea36dc5b178bdfc89e8", "textHash": "9f59fab0f3b3152b54b5de6095911a097d100760f359af89e57dae6dae4f3140", "characters": 1434, "flags": [], "candidateHeadings": [ "8.1 Introduction", "8.2 Counting in Bundles" ], "candidateMarkers": [ { "id": "8.1", "label": null, "offset": 105 }, { "id": "8.2", "label": null, "offset": 560 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0156.png", "sha256": "324dcb086fa4dc27d1c7a96a476a6cdfd185cc034aceb8154def52a443c71d01", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 157, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "88e776d8c236dd157b5762d01b96b566c3802f9a765afad85dcf349971afe415", "textHash": "933c8329a1c644252b5b4eaf556b22859602966d1879565e456dadcd2f86815a", "characters": 623, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.1", "label": "Problem", "offset": 24 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0157.png", "sha256": "020605e8d9d2d8ff0223095080782cc30f19e2ce1329a07bf2e5811220ff2a16", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 158, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4eb53d1870307c5707a8cc0e7cf41168df132bbce78243b100d48c273e77c6d6", "textHash": "4371b155ccabccbe6d3a5d93a1b810284d5d1ce06137de3ceb9d07b27a430bf2", "characters": 7661, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [ "Concept: It's easier to count a large number of objects when we group those objects as we count them." ], "candidateMarkers": [ { "id": "8.1", "label": "Problem", "offset": 25 }, { "id": "8.1", "label": "Solution for Problem", "offset": 144 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0158.png", "sha256": "ba258bf67cb6f7fde708e657099be5b6c3ba215f5e0332584a26b342f55eac10", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 159, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1d98eed62b2a9c940a86cce1d353a01133b67f4a8be34aced476e13797cc1fef", "textHash": "4290eab68c86c1811f479ddb509683373a22ad7ce4e9660c8f40a4805b30964b", "characters": 2196, "flags": [], "candidateHeadings": [ "Definition: A numeral is a symbol used to represent a number.", "Definition: Systems for writing numbers using a select list of digits to represent digit bundles are known as base number systems.", "Exercises" ], "candidateMarkers": [ { "id": "8.2.1", "label": null, "offset": 1470 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0159.png", "sha256": "2bc741f88953140d0bc1b83114ff8b73f8475b586f9a73171c1d8df2323b1a17", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 160, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a00f7d05a00663dd24cb48e1284ead76d4d186b373104fb1a938efdb33e304e3", "textHash": "6e94634a416d2bf6b1696e20b52b566346dc47a123228ed99d315410c2954179", "characters": 1149, "flags": [], "candidateHeadings": [ "8.3 Base Numbers" ], "candidateMarkers": [ { "id": "8.3", "label": null, "offset": 0 }, { "id": "8.2", "label": "Problem", "offset": 112 }, { "id": "8.3", "label": "Problem", "offset": 456 }, { "id": "8.2", "label": "Problem", "offset": 700 }, { "id": "8.2", "label": "Solution for Problem", "offset": 1044 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0160.png", "sha256": "7401d238c388c49eed231a318176a7e23a31a721692598467c2af988f12bccd8", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 161, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "01fb26bb487649b2449ba24d294337a53c5fff9ec223d20379da99f66705bf8a", "textHash": "988a6371f3b45f1e596d9c66b0025495a41557338bdc3df2149997e9c1501942", "characters": 1833, "flags": [], "candidateHeadings": [ "Definition: When we write numerals using digits that represent the first \\(b\\) whole numbers \\((0,1,2,\\ldots,b-2,b-1)\\), we say that we are writing numbers in base \\(b\\). We call \\(b\\) the base of the number system. The base of a number system is also sometimes called the radix or scale.", "Concept: In the decimal system, we write numerals based on bundles of 10. In base 6, we write numerals based on bundles of 6. In base \\(b\\), we write numerals based on bundles of \\(b\\)." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0161.png", "sha256": "4d64e573df7a77abba611f9c0a5cbc70e2e3a210807a5e5bfa412d1b012a1c38", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 162, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3d064b6c996d10e6c706760f8325fffdf493fdb0030a7d0bb687cd45d29f9700", "textHash": "7a596c3406edd24da8ec03f7d9626d62208993cde98adc8e39cc0c94791b99c7", "characters": 1909, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "8.3", "label": "Problem", "offset": 774 }, { "id": "8.3", "label": "Solution for Problem", "offset": 838 }, { "id": "8.3.1", "label": null, "offset": 1305 }, { "id": "8.3.2", "label": null, "offset": 1556 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0162.png", "sha256": "02f6506925886e9e7d59ddb28b7fc4c0351c5ae029f62c4ccbeec416c4db2492", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 163, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "eb19d8f35155f208d0ef4c89a10c7cab1d5da42f676c350ea635bf0d9ced800f", "textHash": "7eb08abd751214c2fe4d7824ccfb9d92362972397e97604a676ba30bafb652fe", "characters": 1149, "flags": [], "candidateHeadings": [ "8.4 Base Number Digits" ], "candidateMarkers": [ { "id": "8.3.3", "label": null, "offset": 24 }, { "id": "8.3.4", "label": null, "offset": 78 }, { "id": "8.3.5", "label": null, "offset": 131 }, { "id": "8.3.6", "label": null, "offset": 184 }, { "id": "8.4", "label": null, "offset": 464 }, { "id": "8.4", "label": "Problem", "offset": 682 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0163.png", "sha256": "3dfd85159d05853600d9dcf039e45f69610b70347ee4ada484c9fc005fd09ddd", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 164, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5b00dc05c11dbd2c500dd81039317ea9b5ae2623654dba8ab6c2132f36b0fe53", "textHash": "7363607be133db0a43e1da8d1c73255cf2699763dce5ea5c0f1842b12028f0b2", "characters": 2334, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "8.4", "label": "Problem", "offset": 24 }, { "id": "8.4", "label": "Solution for Problem", "offset": 92 }, { "id": "8.4.1", "label": null, "offset": 2006 }, { "id": "8.4.2", "label": null, "offset": 2061 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0164.png", "sha256": "6a2a933b169bf82dff5f466b9bcf168c0763d96e92a1c5d296d662d7a097b35a", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 165, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9e0bffe18ce846a582b1bdcff8bec31bfdbecc89fadf6f2cebbb0f5170942138", "textHash": "909703a53722b91eb2ce4251b854fc766656b4492aa8020f9946c26562d31ea5", "characters": 1750, "flags": [], "candidateHeadings": [ "8.5 Converting Integers Between Bases" ], "candidateMarkers": [ { "id": "8.5", "label": null, "offset": 24 }, { "id": "8.5", "label": "Problem", "offset": 73 }, { "id": "8.6", "label": "Problem", "offset": 200 }, { "id": "8.7", "label": "Problem", "offset": 727 }, { "id": "8.8", "label": "Problem", "offset": 1243 }, { "id": "8.9", "label": "Problem", "offset": 1330 }, { "id": "8.10", "label": "Problem", "offset": 1434 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0165.png", "sha256": "4b8629085b018c7beb0e0fa071d4c5c8eef0fc2fc77e0753a8dfad48e456102f", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 166, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5be8c86ec04de6405c7c6a11ceaaf8f4a065451a962c2cdbb404a5f82767f21f", "textHash": "db1013b69bb54df918a0be74c93ccba9d427bdd4870de3ad50216b9c03631f09", "characters": 1320, "flags": [], "candidateHeadings": [ "Important: We convert a number from base b to decimal by adding the decimal values of the number's digit bundles." ], "candidateMarkers": [ { "id": "8.5", "label": "Problem", "offset": 39 }, { "id": "8.5", "label": "Solution for Problem", "offset": 78 }, { "id": "8.6", "label": "Problem", "offset": 1027 }, { "id": "8.6", "label": "Solution for Problem", "offset": 1083 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0166.png", "sha256": "950b1cdcf9e39099eea81ddb4198b1e3bba876fb3732dfabf3b6f669564b482a", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 167, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8ade983f4cf8a37294a1f1ccef54cdfe3b93651945cc0a6451f541b27a19a954", "textHash": "59e44252a5cd656930c414ede4b5b5e15c3a3244ad786778f7b2ef23c0ec5fc8", "characters": 1565, "flags": [], "candidateHeadings": [ "Important: We convert a decimal number to a base b by finding a way to write the number as a sum of base b digit bundles. The coefficients of the digit bundles are the digits of the base b number (including 0 coefficients)." ], "candidateMarkers": [ { "id": "8.7", "label": "Problem", "offset": 1507 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0167.png", "sha256": "f81efce202ead6ac41d5b454578ad228374bedbbbdade9012ea624b9e7e74a43", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 168, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ee94cb901ebdb51cd540bdedfb39c735241cbea20977ac8ce0f509d66d2af1be", "textHash": "40b0c0c0af1fe0ee160821b142b9edca0aed70796d202383f2fe15e37e3ad7e6", "characters": 1784, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.7", "label": "Solution for Problem", "offset": 39 }, { "id": "8.8", "label": "Problem", "offset": 1509 }, { "id": "8.8", "label": "Solution for Problem", "offset": 1547 }, { "id": "8.9", "label": "Problem", "offset": 1741 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0168.png", "sha256": "640ae1d876775fc7e90466ce937e47dcc30b757224e0e28732a050057bdb0b5d", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 169, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b13701ca22d7ca1008879d0eeb04790db09b8aec5b440d048f42ba593037cae2", "textHash": "bab764bf45f40d1d8f51f9bea5b0d70a3f86ddc183f4ce49f0c6aac023fdda8a", "characters": 1676, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "8.9", "label": "Solution for Problem", "offset": 24 }, { "id": "8.10", "label": "Problem", "offset": 290 }, { "id": "8.10", "label": "Solution for Problem", "offset": 393 }, { "id": "8.5.1", "label": null, "offset": 979 }, { "id": "8.5.2", "label": null, "offset": 1143 }, { "id": "8.5.3", "label": null, "offset": 1255 }, { "id": "8.5.4", "label": null, "offset": 1558 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0169.png", "sha256": "d50511e8b368906a806da004f4a47cc1b3c6a4f379fadf877d304b31bf359edb", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 170, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "60b5d288eb5a069b8fb6dd00f5fc2ece1f3171fb9b71da0a5b0886fe442b5fb7", "textHash": "18d2db178f762f7c9ddc7ad2dee3eb1a51236dd4807c1893b5fe9d5f2bbc920c", "characters": 1637, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.6", "label": null, "offset": 0 }, { "id": "8.6", "label": null, "offset": 35 }, { "id": "8.11", "label": "Problem", "offset": 315 }, { "id": "8.12", "label": "Problem", "offset": 539 }, { "id": "8.13", "label": "Problem", "offset": 955 }, { "id": "8.14", "label": "Problem", "offset": 1298 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0170.png", "sha256": "31930e8395b3348da3949290d8b8564eb2f05ad0086b7c6b302f939f915eae92", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 171, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4836af5c0df5f47cc343a65bdd600759d0630a251b19387f07332a7635a4f735", "textHash": "108ddd52cd10616bfb3cb6a1f1c3464035d70456c03f2cb56311f61a15204441", "characters": 1951, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.15", "label": "Problem", "offset": 24 }, { "id": "8.16", "label": "Problem", "offset": 368 }, { "id": "8.11", "label": "Problem", "offset": 629 }, { "id": "8.11", "label": "Solution for Problem", "offset": 715 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0171.png", "sha256": "df49c6f068e995a8153615eb64b4d81c680f125105a3d08692d2af66abb36a15", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 172, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9da8023b930688dde95fe9f19ed28128f9c17670ef4c87d4e053b7537fe6d33b", "textHash": "73d8b3e661a684335e949d3e028bdd2bce728ee9bb45bc33d834e358e6df7574", "characters": 2832, "flags": [], "candidateHeadings": [ "Concept: Solving base number problems often requires a full understanding of the ways in which base numerals are constructed from the numbers they represent." ], "candidateMarkers": [ { "id": "8.6", "label": null, "offset": 0 }, { "id": "8.12", "label": "Problem", "offset": 1009 }, { "id": "8.12", "label": "Solution for Problem", "offset": 1080 }, { "id": "8.13", "label": "Problem", "offset": 2398 }, { "id": "8.13", "label": "Solution for Problem", "offset": 2479 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0172.png", "sha256": "50271e44d3ce33baea764674098e4a37342973d34d44feae45735e40a8874f0b", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 173, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7c0c766dcd882bda83265eb8a9719b5e157edc9f69de6848df4609fe6a839798", "textHash": "38abb2bd4ce3b76a67dcd6482cb37851cfa0d7f723e621d620307fde28881ec8", "characters": 2495, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.14", "label": "Problem", "offset": 2422 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0173.png", "sha256": "9dea348b8b4041be3c78380ab7882994b656f87f2f5bc835dfff2538318d9303", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 174, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "af80095ee59002d055146f4d4ea4e0cab1c9b7b28b528769ebef1fbc7366b4bd", "textHash": "29a5765d978c504503a124044cb32f435c03bec697a5af0a7b837eb30e026c46", "characters": 2291, "flags": [], "candidateHeadings": [ "Concept: Algebraic methods sometimes make problems involving base numbers easier." ], "candidateMarkers": [ { "id": "8.6", "label": null, "offset": 0 }, { "id": "8.14", "label": "Solution for Problem", "offset": 35 }, { "id": "8.15", "label": "Problem", "offset": 641 }, { "id": "8.15", "label": "Solution for Problem", "offset": 724 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0174.png", "sha256": "ffd9ec341f0c62a047ea191471c522b635dc37f68d7a3fa960c06ed64cb602a0", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 175, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "03f182083b84036db48356f3271ce86bf691d760d04e5464ffcb4f95cd33ab7b", "textHash": "dca493b42c9d6140082246d86ffedbadf1afefa90051669c03fdec4e25f431a7", "characters": 1580, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "8.16", "label": "Problem", "offset": 24 }, { "id": "8.16", "label": "Solution for Problem", "offset": 147 }, { "id": "8.6.1", "label": null, "offset": 643 }, { "id": "8.6.2", "label": null, "offset": 891 }, { "id": "8.6.3", "label": null, "offset": 1154 }, { "id": "8.6.4", "label": null, "offset": 1332 }, { "id": "8.6.5", "label": null, "offset": 1460 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0175.png", "sha256": "43c5ccaef3a4282a47de21bbe551da32f0a181fb4fb19fa60113e79c7873da4a", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 176, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1d839f6b442dc7e5e5a970fdf2e94c42725b5fc556b0cd13cecff26cdbe2b21a", "textHash": "665ee54c82018b283111bf75d6aa39cc8fdbde940f10ed39fb3afd8a34b648d3", "characters": 2053, "flags": [], "candidateHeadings": [ "8.7 Summary", "Definitions:", "Concept: In the decimal system, we write numerals based on bundles of 10. In base 6, we write numerals based on bundles of 6. In base \\(b\\), we write numerals based on bundles of \\(b\\).", "Important: When converting a number to a base, we must take note of the coefficients to digit bundles that are 0.", "Concept: Solving base number problems often requires a full understanding of the ways in which base numerals are constructed from the numbers they represent.", "Concept: Algebraic methods sometimes make problems involving base numbers easier." ], "candidateMarkers": [ { "id": "8.7", "label": null, "offset": 13 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0176.png", "sha256": "fb4942a153960c6444442db8b0dbc87a6db4276ecf6d3648fb918985b69a6522", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 177, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5f910fe3ea61ac256beec8ebd371ed6c5f1312e5c3c187770e8e9ec617dd01e6", "textHash": "078258a6ac7880916cd9728100d43fa4be3fa22c79b736573830eebe5d8a30b5", "characters": 1422, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "8.17 Write each of the following as a single base 3 numeral.", "8.18 Count up through the first twenty natural numbers in base 4.", "8.19 Convert each of the following to a base 10 numeral.", "8.20 Convert each of the following decimal integers to the indicated base.", "8.21 Convert each of the following to binary.", "8.22 Perform the following number base conversions.", "8.23 What is the smallest whole number \\( b \\) such that 62 can be expressed in base \\( b \\) using only three digits?" ], "candidateMarkers": [ { "id": "8.17", "label": null, "offset": 388 }, { "id": "8.18", "label": null, "offset": 670 }, { "id": "8.19", "label": null, "offset": 737 }, { "id": "8.20", "label": null, "offset": 885 }, { "id": "8.21", "label": null, "offset": 1033 }, { "id": "8.22", "label": null, "offset": 1099 }, { "id": "8.23", "label": null, "offset": 1299 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0177.png", "sha256": "237fcfde14be8ae15a78f22acf32cf96c0313ba516e12170fc66418297193922", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 178, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "550fa78f9946c2fac75be51cab239cdb45ff752518a6329bd1ba0d5d760b5b17", "textHash": "184aacd4d35506d276efc935a750dc0b1c3c2080134bc117fb4684d241909bbb", "characters": 2255, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "Challenge Problems", "8.24 The number 10!, when written in the base 12 system, ends with exactly how many zeros? (Source: AHSME) Hints: 85", "8.25 Find the one hundredth smallest positive integer that can be written using no digits other than digits 0 and 1 in base 3. Hints: 138", "8.26 How many of the 512 smallest natural numbers include 7 as a digit when written in base 8? Hints: 59", "8.27 Find the largest decimal integer than can be expressed using 3 digits in each of the following bases. Hints: 12", "8.28 Determine the largest possible integer \\(n\\) such that 942! is divisible by 15\".", "8.29 Is there any natural number \\(b\\) such that 422\\(_{b}\\) is prime? Hints: 100", "8.30 The base three representation of \\(x\\) is 12112211122211112222. Find the first digit (on the left) of the base nine representation of \\(x\\). (Source: AHSME) Hints: 144", "8.31 What is the smallest integer \\(b > 3\\) for which the base \\(b\\) number 23\\(_{b}\\) is a perfect square? (Source: Mandelbrot) Hints: 93", "8.32 An eight-bit binary word is a sequence of eight digits each of which is either 0 or 1. Find the number of eight-bit binary words. (Source: UNCC) Hints: 54", "8.33 Given that \\(8^{5} = 32768\\), how would you write 32767 in base 8? Hints: 131", "8.34 How many natural numbers require 3 digits when written in base 12, but require 4 digits when written in base 8? Hints: 3", "8.36 The numeral 47 in base \\(a\\) represents the same number as 74 in base \\(b\\). Assuming that both bases are positive integers, find the least possible value for \\(a + b\\) and leave your answer in base 10. (Source: AHSME) Hints: 19", "8.37 How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit? Hints: 106" ], "candidateMarkers": [ { "id": "8.24", "label": null, "offset": 39 }, { "id": "8.25", "label": null, "offset": 157 }, { "id": "8.26", "label": null, "offset": 296 }, { "id": "8.27", "label": null, "offset": 402 }, { "id": "8.28", "label": null, "offset": 546 }, { "id": "8.29", "label": null, "offset": 633 }, { "id": "8.30", "label": null, "offset": 716 }, { "id": "8.31", "label": null, "offset": 890 }, { "id": "8.32", "label": null, "offset": 1030 }, { "id": "8.33", "label": null, "offset": 1191 }, { "id": "8.34", "label": null, "offset": 1275 }, { "id": "8.35", "label": null, "offset": 1402 }, { "id": "8.36", "label": null, "offset": 1538 }, { "id": "8.37", "label": null, "offset": 1773 }, { "id": "8.38", "label": null, "offset": 1888 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0178.png", "sha256": "7b63bc12bdae4bfeabbc77a6980952978553b77b7e16268717b383b39616ee8c", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 179, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "07465d228c8c69d22704c33b25be81e355346ba337a55acadd5171a7d77e798c", "textHash": "4f6a629b3caa7203222ccee1fd755b636c86a42c423970606399d95db1345bd2", "characters": 1920, "flags": [], "candidateHeadings": [ "8.40 Perform each of the following base number conversions and try to explain the results." ], "candidateMarkers": [ { "id": "8.39", "label": null, "offset": 24 }, { "id": "8.40", "label": null, "offset": 219 }, { "id": "8.41", "label": null, "offset": 471 }, { "id": "8.42", "label": null, "offset": 777 }, { "id": "8.43", "label": null, "offset": 1067 }, { "id": "8.44", "label": null, "offset": 1573 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/page-0179.png", "sha256": "68728cf6b5f54e17f2effcc59153244d694b5790fe88af610078dd39a805879d", "nextChapterHeadingPDF": 180 } }, { "source": "book-05", "pdfPage": 180, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "725f13adee420db3a34847a49bcd158c25b079fb46d66371016489723530da3b", "textHash": "5acaa600962155de7d1d072aad0ad5477720d0a09f3af9eaa7c1d999a4444d9b", "characters": 1192, "flags": [], "candidateHeadings": [ "9.1 Introduction", "9.2 Base Number Addition" ], "candidateMarkers": [ { "id": "9.1", "label": null, "offset": 153 }, { "id": "9.2", "label": null, "offset": 373 }, { "id": "9.1", "label": "Problem", "offset": 660 }, { "id": "9.2", "label": "Problem", "offset": 923 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0180.png", "sha256": "77d3bf0c2ec56202030a2d70f653c580edaaf6ad70e0cec764ab8690f3e50a12", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 181, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f74c9aea55ddc1557bf5584b7af4cb93d59a2da217cf0fe8cb4b2d87a8c3ee1e", "textHash": "5e3683af82357b4db7bb890f0b23fbb9783decc276742ead0598892c5f322b30", "characters": 1801, "flags": [], "candidateHeadings": [ "Concept: Addition, subtraction, multiplication, and exponentiation are just methods of fast counting. Since base number systems provide us with ways to write counting and arithmetic results, we can use arithmetic in all base number systems." ], "candidateMarkers": [ { "id": "9.3", "label": "Problem", "offset": 34 }, { "id": "9.1", "label": "Problem", "offset": 326 }, { "id": "9.1", "label": "Solution for Problem", "offset": 383 }, { "id": "9.2", "label": "Problem", "offset": 1111 }, { "id": "9.2", "label": "Solution for Problem", "offset": 1170 }, { "id": "9.3", "label": "Problem", "offset": 1608 }, { "id": "9.3", "label": "Solution for Problem", "offset": 1666 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0181.png", "sha256": "b5e0a0ce02bb9e75b8bec41ef4d037c636315c5b5b61096a276c5aadeececc6a", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 182, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c30227ff8e56c3c6f2af1071f53bf1cd16d78dc4baa22a5a7b07421cb717a02e", "textHash": "da88ffae31353140866116455b1dfa9ca7548f80659767cc99ab6eeafdcb5a18", "characters": 1826, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "9.2.1", "label": null, "offset": 1416 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0182.png", "sha256": "37c958d2af0d1ca84596aa82cea483e9a2b97b5c44f3a089f407727ce516fae2", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 183, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e714a0370540e5f4c5cfdb59a262303cd73080194b4296dfa0ab4be2ba13e514", "textHash": "97d476a74302055b1450c50440b8d3782bbdd97ae20b934d23f8e6d12cefa3c6", "characters": 1563, "flags": [], "candidateHeadings": [ "9.3 Base Number Subtraction" ], "candidateMarkers": [ { "id": "9.3", "label": null, "offset": 34 }, { "id": "9.4", "label": "Problem", "offset": 209 }, { "id": "9.5", "label": "Problem", "offset": 409 }, { "id": "9.6", "label": "Problem", "offset": 801 }, { "id": "9.4", "label": "Problem", "offset": 1112 }, { "id": "9.4", "label": "Solution for Problem", "offset": 1170 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0183.png", "sha256": "4365c9fe7e4c14df30330c777733067aff84a6c80b21b5338bf58af3b767bdff", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 184, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d375a6b392f0df12469c59dbafc2921e25ae36d2dc71cb95ecda7f064f10f76c", "textHash": "0af716ce1cf0569e6f18e922085786b95370efccdf2006501b27f5ae90edcee4", "characters": 1582, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "9.5", "label": "Problem", "offset": 29 }, { "id": "9.5", "label": "Solution for Problem", "offset": 74 }, { "id": "9.6", "label": "Problem", "offset": 500 }, { "id": "9.6", "label": "Solution for Problem", "offset": 535 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0184.png", "sha256": "74ec6d1d8ae7ea8e5ad58eff869838a25b85c9472036a0163921d2737d9ce4c3", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 185, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "840d3807797153f0d607b4bb01eefa4d3b293876eff7b406e34a16ddce5867e3", "textHash": "3a76c6418d0745d2708c81af7e75e0cabb1df80a1ed7ecbe5cd93acf82ee09a5", "characters": 1612, "flags": [], "candidateHeadings": [ "Exercises", "9.4 Base Number Multiplication" ], "candidateMarkers": [ { "id": "9.3.1", "label": null, "offset": 339 }, { "id": "9.4", "label": null, "offset": 637 }, { "id": "9.7", "label": "Problem", "offset": 1019 }, { "id": "9.8", "label": "Problem", "offset": 1091 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0185.png", "sha256": "74d07b47f1c60b3323da0cb46812bf0c0f0cd3bace1419d6f7a23b34cc2be853", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 186, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b495be14900c15fbfe8969b72013c5f6ad1a77bd237759bd0e51c2e142c556d5", "textHash": "6b0df6522798525be43800a5684a698de4c14caa35e2d5fcccd0921fc197a093", "characters": 2463, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "9.7", "label": "Problem", "offset": 32 }, { "id": "9.7", "label": "Solution for Problem", "offset": 87 }, { "id": "9.8", "label": "Problem", "offset": 469 }, { "id": "9.8", "label": "Solution for Problem", "offset": 559 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0186.png", "sha256": "68daf8d31492ba8cf9364198be4968c99a0112175a3a8f2aef90ccec0d89d020", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 187, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6b892b469e98e344be032e313ec5fe353a3a60f0026bbb34e765a9e811f81abe", "textHash": "b3c0e7994b027011565145a331859e085943d9735fde630137b19b6b44776313", "characters": 935, "flags": [], "candidateHeadings": [ "9.5 Base Number Division and Divisibility" ], "candidateMarkers": [ { "id": "9.4.1", "label": null, "offset": 269 }, { "id": "9.5", "label": null, "offset": 463 }, { "id": "9.9", "label": "Problem", "offset": 514 }, { "id": "9.10", "label": "Problem", "offset": 812 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0187.png", "sha256": "6e169b80572af4c1e96f3c59de4af14f696a3ea21294822a06bcb2ca20dc735b", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 188, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "af396b5cd5d20d50db4f57f14ee13e506c2137f587f75356117847d8bc5842c4", "textHash": "29a1961790e70532eeb5fcdb2c88a583952edc7f8a6df46275dc26cc40c2513d", "characters": 1928, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "9.9", "label": "Problem", "offset": 43 }, { "id": "9.9", "label": "Solution for Problem", "offset": 100 }, { "id": "9.10", "label": "Problem", "offset": 876 }, { "id": "9.10", "label": "Solution for Problem", "offset": 1030 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0188.png", "sha256": "0e35873a1eb0e7d45667cce671ea0047578cc9134d2e90f1fb86b4147a3db1ff", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 189, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ee7db102e74e199fe937dbabdf533890cf6814278a2df942b3654c885be016a7", "textHash": "f4f31e3247f3052d22b0c840520fefaac71fc09e380d878ebc811b92a9510b3b", "characters": 1943, "flags": [], "candidateHeadings": [ "5.72 + 2.72 + 4.71 + 1.70 = 7.72 + 4.71 + 1.70", "7.71 + 4.71 + 1.70 = 11.71 + 1.70", "Exercises" ], "candidateMarkers": [ { "id": "5.72", "label": null, "offset": 513 }, { "id": "7.71", "label": null, "offset": 765 }, { "id": "9.5.1", "label": null, "offset": 1569 }, { "id": "9.5.2", "label": null, "offset": 1758 }, { "id": "9.5.3", "label": null, "offset": 1792 }, { "id": "9.5.4", "label": null, "offset": 1826 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0189.png", "sha256": "4d579cdb3283955d34b31ad0ae9e753dfb5faa119bfe52d0c8dd6ab9361b837c", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 190, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4fa69adc7c05b94fbe7162e68c9755508ade6e7e0952d22458715ea3235609f0", "textHash": "f363a79cbfcf38364c833163db9a21f041b06b97976901c4a154c3b3a5a43552", "characters": 1244, "flags": [], "candidateHeadings": [ "9.6 Summary", "Concept: Addition, subtraction, multiplication, and exponentiation are just methods of fast counting. Since base number systems provide us with ways to write counting and arithmetic results, we can use arithmetic in all base number systems.", "REVIEW PROBLEMS", "9.11 Perform the indicated addition.", "9.12 Perform the indicated subtraction.", "9.13 Perform the indicated multiplication.", "9.14 Perform the indicated division.", "9.15 Determine which of the following are multiples of 5.", "9.16 Determine which of the following are multiples of 12." ], "candidateMarkers": [ { "id": "9.6", "label": null, "offset": 13 }, { "id": "9.11", "label": null, "offset": 578 }, { "id": "9.12", "label": null, "offset": 693 }, { "id": "9.13", "label": null, "offset": 781 }, { "id": "9.14", "label": null, "offset": 876 }, { "id": "9.15", "label": null, "offset": 960 }, { "id": "9.16", "label": null, "offset": 1100 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0190.png", "sha256": "059dcedfc5cbd2840d7a65914a20c8b3476149893545bec2ee14f3957f820797", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 191, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ec2aa389ef649afca1ae9e8abaa42a16472bfb275620cdf5d4899397434a812b", "textHash": "76b6192d8e751fe02167e332548ce82606bd0b471f241836b5e88d61364210fc", "characters": 2123, "flags": [], "candidateHeadings": [ "Challenge Problems", "9.17 Find the value of the base \\( b \\) such that the following addition problem is correct:", "9.18 Compute: (101012 + 10112) ÷ (1100112 + 11012) ÷ (10002 + 1002 + 102 + 12 + 12). Hints: 82", "9.19 Find the positive base \\( b \\) in which the equation \\( 4 \\cdot 12 = 103 \\) is valid. Hints: 141", "9.20 Is there any base \\( b \\) for which 3443\\(_{b}\\) is prime? If so, provide an example. If not, explain why not. Hints: 35", "9.21 Find the largest prime number (in decimal form) that divides the sum,", "9.22 A binary number consists of 17 digits, all of which are 1. Triple the number.", "9.23 Let the product (12)(15)(16), each factor written in base \\( b \\), equal 3146 in base \\( b \\). Let \\( s = 12 + 15 + 16 \\), each term expressed in base \\( b \\). Find the value of \\( s \\) in base \\( b \\). (Source: AHSME) Hints: 51", "9.24 Cleo needs to determine if a 71777-digit base 12 integer is a multiple of 3. The last five digits of the integer are 71777. Find a method that helps Cleo quickly determine whether or not the enormous integer is a multiple of 3. Hints: 27", "9.25 The evil villain Harris Pilton wrote three secret two-digit numbers, \\( x \\), \\( y \\), and \\( z \\) on a napkin. Berris Fueler must name three numbers, \\( A \\), \\( B \\), and \\( C \\), after which Harris will announce the value of \\( Ax + By + Cz \\). If Berris can then name Harris' three secret numbers, Harris will let Berris go. Save Berris! Come up with a way to choose \\( A \\), \\( B \\), and \\( C \\) such that Berris is sure to escape Harris' evil clutches. Hints: 96", "9.26 Find the sum of all the natural numbers that are three-digit palindromes when expressed in base 5. Express your answer in base 5. Hints: 149" ], "candidateMarkers": [ { "id": "9.17", "label": null, "offset": 54 }, { "id": "9.18", "label": null, "offset": 190 }, { "id": "9.19", "label": null, "offset": 286 }, { "id": "9.20", "label": null, "offset": 389 }, { "id": "9.21", "label": null, "offset": 516 }, { "id": "9.22", "label": null, "offset": 649 }, { "id": "9.23", "label": null, "offset": 841 }, { "id": "9.24", "label": null, "offset": 1076 }, { "id": "9.25", "label": null, "offset": 1320 }, { "id": "9.26", "label": null, "offset": 1795 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/page-0191.png", "sha256": "688dae295961b71d9ec5b253661f1c888ea8798c47a919a13d45fe188123a8ad", "nextChapterHeadingPDF": 192 } }, { "source": "book-05", "pdfPage": 192, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5300798486efa732289ff92aa4d83eb84cfd765409cd0545c0ea4f319b280e46", "textHash": "42ea61443ea90869db7d1230d7a2a7833ed8213ab6c748284f0fa996a347a86e", "characters": 1203, "flags": [], "candidateHeadings": [ "10.1 Introduction", "10.2 Units Digits in Arithmetic" ], "candidateMarkers": [ { "id": "10.1", "label": null, "offset": 100 }, { "id": "10.2", "label": null, "offset": 603 }, { "id": "10.1", "label": "Problem", "offset": 908 }, { "id": "10.2", "label": "Problem", "offset": 1065 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0192.png", "sha256": "8358f4638902f13070fc498ee7c59098dba79cca6c68a0c221dbc020b0065bcf", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 193, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e6039791c60ee3a5b1e94395a1027e34da62faef521dfbb9b6487a0d9ae6e868", "textHash": "9407f59fd7e20bc24bbc2a8c801022f7f124ad3fef1179ad6e06535dd310ba05", "characters": 1899, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.3", "label": "Problem", "offset": 25 }, { "id": "10.4", "label": "Problem", "offset": 192 }, { "id": "10.5", "label": "Problem", "offset": 315 }, { "id": "10.6", "label": "Problem", "offset": 470 }, { "id": "10.7", "label": "Problem", "offset": 615 }, { "id": "10.8", "label": "Problem", "offset": 780 }, { "id": "10.9", "label": "Problem", "offset": 946 }, { "id": "10.1", "label": "Problem", "offset": 1185 }, { "id": "10.1", "label": "Solution for Problem", "offset": 1343 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0193.png", "sha256": "356a6114404e126402d477559a3c28a3750b14520d0cb0fe8c1eb5faa8aaea70", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 194, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1fce3a84deb57866068e757988463f1190066d114ea44634e0632998b3893753", "textHash": "41656f4ee3f0f4c3e1e0ad57dfd34e0f2d12d94c2f0de905475334b2b3742603", "characters": 2365, "flags": [], "candidateHeadings": [ "Important: The units digit of the sum of two positive integers is the same as the units digit of the sum of just their units digits. Repeating this process, the units digit of the sum of a group of positive integers is the same as the units digit of the sum of just the units digits of those positive integers." ], "candidateMarkers": [ { "id": "10.2", "label": "Problem", "offset": 1686 }, { "id": "10.2", "label": "Solution for Problem", "offset": 1821 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0194.png", "sha256": "d4a32eb888a66d1496e56d8873b3c03f05ef52ca6ab28964f8a21579a36660c1", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 195, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "53bd31066ec3d5de837b72c6ff9cd3db8a8984844ac9a42d855514c56496064a", "textHash": "1d159def2b7c8eede06967883bd35a3ab77f20a934d69da6c3606b123ec4d727", "characters": 1674, "flags": [], "candidateHeadings": [ "Important: When we subtract a positive integer from a larger integer, the units digit of the result equals the result of subtracting the units digit of the smaller integer from the units digit of the larger (carrying if necessary). This result can be extended to subtraction of several integers at a time." ], "candidateMarkers": [ { "id": "10.3", "label": "Problem", "offset": 25 }, { "id": "10.3", "label": "Solution for Problem", "offset": 466 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0195.png", "sha256": "b7b5d3a3176822ec8320a22227115bf4344ed14a93d34e0dd4d03c23f54f4522", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 196, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "24d10be94a3fdc28b23c67810a45a320e4a3ab94f233e27af4995cc6454c2305", "textHash": "b8f9779e857659474410cd0e387cce6de6c607cad18503902741c881b56e25f0", "characters": 9578, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.4", "label": "Problem", "offset": 33 }, { "id": "10.4", "label": "Solution for Problem", "offset": 157 }, { "id": "10.5", "label": "Problem", "offset": 789 }, { "id": "10.5", "label": "Solution for Problem", "offset": 944 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0196.png", "sha256": "1a2546878a13ad26619353a986755e6a58eb33cfb69de722412e6bc13f082bca", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 197, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fa1129d3cb588df16899393a91ede4d58cd603bc77b37268e1e0bd4f454f0baf", "textHash": "e779725d5b741fe9686c1dfc11349136270fb8de1c3f14cc18319687fa4f1af6", "characters": 2020, "flags": [], "candidateHeadings": [ "4.4 = 16", "4.6 = 24", "Concept: Hunting for patterns can be very helpful when solving problems or learning more about mathematics. Play around with numbers and test simple cases. See if you can establish a pattern. Then try to determine why the pattern you discovered works." ], "candidateMarkers": [ { "id": "10.6", "label": "Problem", "offset": 25 }, { "id": "10.6", "label": "Solution for Problem", "offset": 158 }, { "id": "10.7", "label": "Problem", "offset": 596 }, { "id": "10.7", "label": "Solution for Problem", "offset": 642 }, { "id": "4.4", "label": null, "offset": 1356 }, { "id": "4.6", "label": null, "offset": 1365 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0197.png", "sha256": "c7a465d07fc402392a88c7e19394225620baca74905567c9e6b1622799ae9734", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 198, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4391c6071f664efa24bfe67f831296ea4ec79b21c8684636ee3edabd89b77f8e", "textHash": "5962afbbb79fc6bb0a227ad3cc3e57a49fb44df7c1e2c335dcdecb2de55d6bed", "characters": 2058, "flags": [], "candidateHeadings": [ "1.7 = 7", "3.7 = 21", "5.7 = 35", "7.7 = 49", "9.7 = 63", "Concept: The possible number of units digits in a problem is usually very limited. We can solve such problems involving units digits by simply computing the possible outcomes.", "Concept: Translate word problems into the language of mathematics so that you can solve them mathematically." ], "candidateMarkers": [ { "id": "10.8", "label": "Problem", "offset": 33 }, { "id": "10.8", "label": "Solution for Problem", "offset": 196 }, { "id": "1.7", "label": null, "offset": 224 }, { "id": "3.7", "label": null, "offset": 233 }, { "id": "5.7", "label": null, "offset": 242 }, { "id": "7.7", "label": null, "offset": 251 }, { "id": "9.7", "label": null, "offset": 260 }, { "id": "10.9", "label": "Problem", "offset": 763 }, { "id": "10.9", "label": "Solution for Problem", "offset": 1002 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0198.png", "sha256": "8246868919de29021e9feac08c7805b4fe4ea05bc95bafcab8fd5a3ebd87631e", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 199, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "070397415caa03b77a932638e2b1ca05ea2b7212bbfb99bab2310579a07430ef", "textHash": "40a2f6b195d849a770d91b13aa94997634d6859d7a8b03d1338c80a4173dad9d", "characters": 1729, "flags": [], "candidateHeadings": [ "Exercises", "10.3 Base Number Units Digits" ], "candidateMarkers": [ { "id": "10.2.1", "label": null, "offset": 35 }, { "id": "10.2.2", "label": null, "offset": 391 }, { "id": "10.2.3", "label": null, "offset": 462 }, { "id": "10.2.4", "label": null, "offset": 593 }, { "id": "10.2.5", "label": null, "offset": 713 }, { "id": "10.2.6", "label": null, "offset": 882 }, { "id": "10.3", "label": null, "offset": 1133 }, { "id": "10.10", "label": "Problem", "offset": 1466 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0199.png", "sha256": "bd8d7628e3744f922f4e1c4fbc5c6a1faf74e06243c73552003f7fc1994dc7e4", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 200, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ab219f4f521f7e12675cfd4090aaa4aaeb55ac2a1f1e539f57d9e376e94ebfe6", "textHash": "80e36dc815a8931aa180b62cad654a3090b67edc22e28cd3f94c82f73e39db73", "characters": 2100, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.11", "label": "Problem", "offset": 31 }, { "id": "10.12", "label": "Problem", "offset": 161 }, { "id": "10.13", "label": "Problem", "offset": 219 }, { "id": "10.10", "label": "Problem", "offset": 478 }, { "id": "10.10", "label": "Solution for Problem", "offset": 741 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0200.png", "sha256": "8a4eb2a58765bc4bb7466e4efe4c00bc9fff1031de0642299cd0f1baacdfb01b", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 201, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "bd2487a04607f3a773f1743508e020cac43ed9d62d272fd0bdbe011d5bd9285f", "textHash": "ca825d354ba35e5cda4e919ff3a6a6db8c892bebd34c84f1cf22220f2aba3eb6", "characters": 2343, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.11", "label": "Problem", "offset": 25 }, { "id": "10.11", "label": "Solution for Problem", "offset": 155 }, { "id": "10.12", "label": "Problem", "offset": 689 }, { "id": "10.12", "label": "Solution for Problem", "offset": 747 }, { "id": "10.13", "label": "Problem", "offset": 1640 }, { "id": "10.13", "label": "Solution for Problem", "offset": 1899 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0201.png", "sha256": "de136381de0eed947df951628ab54c48784bd4338da7b8d223af96163c45c51d", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 202, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "962acb7a60cf589b2e936b8cbd39c96747caa11e7fc5827f3e03e035815ebda5", "textHash": "78a28dbd08c9e03a9aa663e4aa6989b43c983be9bf1595b5414a42afa9ef9372", "characters": 1894, "flags": [], "candidateHeadings": [ "Exercises", "10.4 Unit Digits Everywhere!" ], "candidateMarkers": [ { "id": "10.3.1", "label": null, "offset": 41 }, { "id": "10.3.2", "label": null, "offset": 362 }, { "id": "10.3.3", "label": null, "offset": 663 }, { "id": "10.4", "label": null, "offset": 829 }, { "id": "10.14", "label": "Problem", "offset": 869 }, { "id": "10.15", "label": "Problem", "offset": 940 }, { "id": "10.16", "label": "Problem", "offset": 1029 }, { "id": "10.17", "label": "Problem", "offset": 1118 }, { "id": "10.18", "label": "Problem", "offset": 1336 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0202.png", "sha256": "69b242734f98db418349dc559091040d3bc66f522e74643b90bc28e86ff18c67", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 203, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1a7efcb8f9fa2f9d5656b7732b8e5c49c8e649bd98b5bc076bfd4c4a1012c6d0", "textHash": "4a0ac1b68bada6bf513da01ec8b2a6f5623d358de409aa1375b2232bc41f658c", "characters": 1443, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.14", "label": "Problem", "offset": 25 }, { "id": "10.14", "label": "Solution for Problem", "offset": 96 }, { "id": "10.15", "label": "Problem", "offset": 272 }, { "id": "10.15", "label": "Solution for Problem", "offset": 361 }, { "id": "10.16", "label": "Problem", "offset": 974 }, { "id": "10.16", "label": "Solution for Problem", "offset": 1063 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0203.png", "sha256": "0cdc9cfc2f7ae76d974876000c2555112b4cd76aa549b3aaac472d79e5740aba", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 204, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "56610546392e27ff89eed3baf9e339b05528486d4f2b230a43c8e634dd8eb9d2", "textHash": "ab55479d586c87eee064f3cef047623ff7b9e07a6aec7b30b0c3bce48715bdf4", "characters": 2262, "flags": [], "candidateHeadings": [ "1.5 = 5", "3.5 = 15", "5.5 = 25", "7.5 = 35", "9.5 = 45" ], "candidateMarkers": [ { "id": "10.17", "label": "Problem", "offset": 30 }, { "id": "10.17", "label": "Solution for Problem", "offset": 248 }, { "id": "1.5", "label": null, "offset": 472 }, { "id": "3.5", "label": null, "offset": 481 }, { "id": "5.5", "label": null, "offset": 490 }, { "id": "7.5", "label": null, "offset": 499 }, { "id": "9.5", "label": null, "offset": 508 }, { "id": "10.18", "label": "Problem", "offset": 940 }, { "id": "10.18", "label": "Solution for Problem", "offset": 1032 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0204.png", "sha256": "bc355e3ba840f81cebb662230337eed244f4bfafb35460ca9319a9b1591d3741", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 205, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9a992a8051b746789136b7a4957a951a0ddaf31b1b304d65067850c76a1d04fa", "textHash": "d432e2289f77b0ee7f91a5986a070cd87dd607ed1a940d058a227bceae2ef021", "characters": 6177, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0205.png", "sha256": "33cdb67b2906050ce37b2711ad0f4bb6097b2eea71f8668bf374c0c633be742f", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 206, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "aabe05420ef034bfb1d6486a8d38c0b5ac52d85c75ba208d7589c9cd70b23562", "textHash": "f35da9f1151818a2999a6a68056a6599f93dcf5251838d9e5e0cc46c38b86c43", "characters": 2107, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Concept: Hunting for patterns can be very helpful when solving problems or learning more about mathematics. Play around with numbers and test simple cases. See if you can establish a pattern. Then try to determine why the pattern you discovered works.", "Concept: Part of the power of focusing on units digits is that they can simplify problems involving many integers into just a few cases that can be easily calculated.", "REVIEW PROBLEMS", "10.19 Find the units digit of each of the following.", "10.20 Find the units digit of 31986 - 21986. (Source: MATHCOUNTS)", "10.21 Jenny has 8 stamp books that each contain 42 pages. Each page in her books contains 6 stamps. Jenny decides to reorganize her stamp books such that each page contains 10 stamps. This will give her more space to collect stamps without having to buy new books. Under her new system, Jenny fills up 4 complete books, still with 42 pages per book. Her fifth book now contains 33 pages filled with 10 stamps per page and 1 last page with the remaining stamps. How many stamps are on that last page?", "10.22 The product of two positive integers has a units digit of 9. If one of the integers has a units digit of 7, what is the units digit of the other?", "10.23 The product of two primes has a units digit of 3. What are the possible units digits of their sum?", "10.24 Find the units digit of each of the following within the indicated number base." ], "candidateMarkers": [ { "id": "10.19", "label": null, "offset": 810 }, { "id": "10.20", "label": null, "offset": 1023 }, { "id": "10.21", "label": null, "offset": 1090 }, { "id": "10.22", "label": null, "offset": 1591 }, { "id": "10.23", "label": null, "offset": 1744 }, { "id": "10.24", "label": null, "offset": 1850 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0206.png", "sha256": "15c4366fc4fec500283fdbc02175461cd6a40d95b5c628204f5487871fbdc5bc", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 207, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f7d5872c29d0312cb18fd009b6d9382ff792a77b2f76006faeaf343e1d492184", "textHash": "e571fe730722670729c58cbe1c3121b913f04caa8b6be016c0fa97de19d087a0", "characters": 1828, "flags": [], "candidateHeadings": [ "10.25 Two positive integers have units digits of 2 and 3 respectively when expressed in base b. Find the units digit of their product when b equals", "10.26 What are the possible units digits of a perfect square written in base 5?", "10.27 What are the possible units digits of a perfect fourth power (in base 10)?", "10.28 If n is a positive integer, what are the possible values of the units digit of \\(\\frac{(n + 2)!}{n!}\\)?", "10.29 When the product of two consecutive positive integers is written in base 7, the units digit is 5. Find the units digit of their base 7 sum.", "Challenge Problems", "10.30 What are the possible units digit of a perfect fourth power written in base 5?", "10.32 What is the units digit of \\((133^{13})^{3}\\)? (Source: MATHCOUNTS)", "10.33 Brannon is reading the book Flatland and notices that the product of the numbers of the two pages his book is open to has a unit digit of 6. What is the unit digit of the sum of the two page numbers? Hints: 101", "10.34 How many of the following integers have a unit digit of 6?", "10.35 Find the units digit of n given that \\(mn = 21^{6}\\) and m has a unit digit of 7.", "10.36 Given that m and n are positive integers such that \\(m^{2}n = 39^{39}\\) and n has units digit 1, find the possible values of the units digit of m. Hints: 90", "10.37 The product of two numbers has been accidentally erased by a student. All that remains is shown here." ], "candidateMarkers": [ { "id": "10.25", "label": null, "offset": 25 }, { "id": "10.26", "label": null, "offset": 204 }, { "id": "10.27", "label": null, "offset": 285 }, { "id": "10.28", "label": null, "offset": 367 }, { "id": "10.29", "label": null, "offset": 478 }, { "id": "10.30", "label": null, "offset": 645 }, { "id": "10.31", "label": null, "offset": 731 }, { "id": "10.32", "label": null, "offset": 1046 }, { "id": "10.33", "label": null, "offset": 1121 }, { "id": "10.34", "label": null, "offset": 1339 }, { "id": "10.35", "label": null, "offset": 1462 }, { "id": "10.36", "label": null, "offset": 1551 }, { "id": "10.37", "label": null, "offset": 1715 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0207.png", "sha256": "2e377b4bbcf8cafe0dbaedd29c555ec01a0f700a8db38e2723466b80d74c236f", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 208, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c15391979ea7285c127d5da611676dff892589a100f9ff8acba0c97e4a148e7f", "textHash": "5001ca323928fc61e471a43a009b559dc36b839d7ab9cd767d3fa7cc91b3abe0", "characters": 1781, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "10.38 How many three-digit integers are multiples of both 4 and 6 and have a unit digit of 2? Hints: 31", "10.39 Let S be the sum of ten consecutive positive integers and let P be their product. Explain why the units digit of P - S must be 5. Hints: 169", "10.40 Find the units digit of \\(3^{200}\\) when expressed in each of the following bases:", "10.41 The cube of the three-digit natural number A7B is 108531333. What is A + B? (Source: MATHCOUNTS) Hints: 4, 108", "10.43 Margaret and her younger sister Wylia are both between 10 and 20 years old. The sum of their ages has a unit digit of 6 and the difference between their ages is 2. If Wylia’s age is an even number, how old is Margaret?", "10.44 Find the smallest integer greater than 2 that has a unit digit of 2 when written in base 3, base 4, base 5, and base 6. (You can leave your answer as a decimal integer.) Hints: 40", "10.45 In how many number bases, b, does", "10.46 How many Mersenne primes have units digit 3? Hints: 163" ], "candidateMarkers": [ { "id": "10.38", "label": null, "offset": 141 }, { "id": "10.39", "label": null, "offset": 246 }, { "id": "10.40", "label": null, "offset": 394 }, { "id": "10.41", "label": null, "offset": 514 }, { "id": "10.42", "label": null, "offset": 632 }, { "id": "10.43", "label": null, "offset": 869 }, { "id": "10.44", "label": null, "offset": 1095 }, { "id": "10.45", "label": null, "offset": 1282 }, { "id": "10.46", "label": null, "offset": 1402 }, { "id": "10.47", "label": null, "offset": 1465 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0208.png", "sha256": "0a4b06647a119917dc6a598a047110704d01d3d8d66cbc593f52c8f48ede4b93", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 209, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b568646a0ccddec2e65abdc603a230af02a76d1280044fc598a6a28c5166f378", "textHash": "0801f49ba7578689f09a7c011716c35408e023abac889fc321564893a4760eba", "characters": 2386, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/page-0209.png", "sha256": "de370f76b7a13600f4c81143d0178ae471e6f5e686d6ecda2d17b67ecf4fe0f7", "nextChapterHeadingPDF": 210 } }, { "source": "book-05", "pdfPage": 210, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "cfc91c0f25349dab1b23e59a4ebddd9bdb98d3e7efa21813cf2de54c3e427a15", "textHash": "27a2f534a47f9749af23cec40773938c484741dcc36ce72e4ace47efb00337dd", "characters": 1425, "flags": [], "candidateHeadings": [ "11.1 Introduction", "11.2 Terminating Decimals", "Definition: A number whose decimal representation ends after a finite number of digits has a terminating decimal expansion. We call these numbers terminating decimals." ], "candidateMarkers": [ { "id": "11.1", "label": null, "offset": 174 }, { "id": "11.2", "label": null, "offset": 649 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0210.png", "sha256": "06f957fe92671c289f78d494449d40259faba2a03d9065299238815ca1aa6840", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 211, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "920eddf5c8f389024b99f574f5198c8bd576364fb5913c6e6dba65c4d8ee7877", "textHash": "159e2f1436d75dd5c14852ca0d9e377aafdd6b6c9e0c3224f7f78ca7a953f0ee", "characters": 1768, "flags": [], "candidateHeadings": [ "0.1 | 0.3 | 1.7 | 2.93 |", "1.0 | 3.0 | 10 | 20 |" ], "candidateMarkers": [ { "id": "11.1", "label": "Problem", "offset": 35 }, { "id": "11.2", "label": "Problem", "offset": 257 }, { "id": "11.3", "label": "Problem", "offset": 448 }, { "id": "11.4", "label": "Problem", "offset": 684 }, { "id": "11.5", "label": "Problem", "offset": 822 }, { "id": "11.1", "label": "Problem", "offset": 1160 }, { "id": "11.1", "label": "Solution for Problem", "offset": 1289 }, { "id": "0.1", "label": null, "offset": 1399 }, { "id": "1.0", "label": null, "offset": 1460 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0211.png", "sha256": "955303306474114c92a3b1ab0998f44f33399ff568ebff4cf8a9c4f4e8843ca1", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 212, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b57b5ab9e4622a8e8a6bd52c85a5555d4aebe5761bcf303fcf6d170f7129a4f2", "textHash": "cb7e33bd295ddbb47d7e5e55dce663c290f6cd3b099fe9654aa5c9061a3d06ee", "characters": 2651, "flags": [], "candidateHeadings": [ "Important: For a positive integer n, when we divide a decimal number by \\(10^n\\), we move the decimal point n places to the left.", "Concept: When faced with a problem involving the properties of numbers or principles of arithmetic, consider whether it's more illuminating to view the numbers in terms of decimals or fractions. The fact that either one of these forms might shed more light on a particular idea than the other is another example of how we use convenient mathematical forms to solve problems." ], "candidateMarkers": [ { "id": "11.2", "label": "Problem", "offset": 1993 }, { "id": "11.2", "label": "Solution for Problem", "offset": 2184 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0212.png", "sha256": "e167b1b363db418de2e4d4eb5c8b72c35d495e84c6487529840bb295c47af685", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 213, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "80ee281bed40e51892c46ac6dbcca55e518d4010fd28e2ae077b21bc0f7b8469", "textHash": "07b1f631ab47d0355a358aebb5538f0323847b62a5b32e3973a66a9825484b6f", "characters": 1979, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.3", "label": "Problem", "offset": 35 }, { "id": "11.3", "label": "Solution for Problem", "offset": 173 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0213.png", "sha256": "a72b164473f80953feef65116f108972b640c99308e5a0f89679f85c40ba2067", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 214, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "55c0a23c9a9c45cb2543e104ce9d8ad5f86d0fa6939a0131fd4950b697af7ea6", "textHash": "aaa3f142fd1f6c23065c1e6ed21faaf802cc47ec6bfadbc8197b27b32efaa58c", "characters": 1922, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.4", "label": "Problem", "offset": 454 }, { "id": "11.4", "label": "Solution for Problem", "offset": 592 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0214.png", "sha256": "73bf6c95858175dd8d40c163876223f69980758c3f257ca458a5f8762a2e74d1", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 215, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "63f92a5e8328af7412fa6db01e7d556209869ca8554d9562a3b8a1a442a63346", "textHash": "a2b6549c4391d2d47319734a26f122f336f0d7b71c4dff997b791ad3fba5ed37", "characters": 1902, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "11.5", "label": "Problem", "offset": 35 }, { "id": "11.5", "label": "Solution for Problem", "offset": 276 }, { "id": "11.2.1", "label": null, "offset": 1611 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0215.png", "sha256": "c55211e7fe64f1f29609508dcfbffe0296cfaa4bd5f17f9c0fcdcf098412172e", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 216, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "423dad8444bd91afb3c2a6ae9bc53286c7713f4c5b5b2f74ce3f9a2a9d3e7b63", "textHash": "7fa9b1a2b9da4ef5683bf02d6773c24b6e2fc523f84241a66e03f0c75331fed0", "characters": 1727, "flags": [], "candidateHeadings": [ "11.3 Repeating Decimals", "Definition: A number whose decimal representation becomes an endlessly repeating sequence of digits has a repeating decimal expansion. We simply call these numbers repeating decimals." ], "candidateMarkers": [ { "id": "11.2.2", "label": null, "offset": 25 }, { "id": "11.3", "label": null, "offset": 183 }, { "id": "11.6", "label": "Problem", "offset": 619 }, { "id": "11.7", "label": "Problem", "offset": 874 }, { "id": "11.8", "label": "Problem", "offset": 1130 }, { "id": "11.9", "label": "Problem", "offset": 1393 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0216.png", "sha256": "57b1b97ad9e21aca44403cad3413cd4c8b372c5ca2660dbb4d73553f58b8b997", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 217, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "57439d638b1031bea419e67f1241bb49d1a16cf4ad6d0ea3d836eb16f3de0877", "textHash": "6a6c768c174dfae3311fc1550d36b49231932313401f61837fbb281e0012173d", "characters": 1725, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.6", "label": "Problem", "offset": 35 }, { "id": "11.6", "label": "Solution for Problem", "offset": 142 }, { "id": "0.9", "label": null, "offset": 729 }, { "id": "0.10", "label": null, "offset": 733 }, { "id": "0.09", "label": null, "offset": 738 }, { "id": "0.01", "label": null, "offset": 743 }, { "id": "11.7", "label": "Problem", "offset": 760 }, { "id": "11.7", "label": "Solution for Problem", "offset": 867 }, { "id": "1.4", "label": null, "offset": 1208 }, { "id": "0.60", "label": null, "offset": 1212 }, { "id": "0.56", "label": null, "offset": 1217 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0217.png", "sha256": "129824cf4423ee453cb8421ebb3470052451949afefee618b6e18aa5ad026cdc", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 218, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3aece43efc7a754d9abae6e20f2e7fe39ab51e49fa12331ff96071c6311aa6fb", "textHash": "9a341c971441c1c8399b0eb2599dc185a8f5ae1600d4fd61b1b6f33503a2b8f3", "characters": 2403, "flags": [], "candidateHeadings": [ "Important: For relatively prime natural numbers \\(a\\) and \\(b\\), the decimal expansion of \\(\\frac{a}{b}\\) terminates if and only if \\(b\\) has no prime divisors other than 2 or 5." ], "candidateMarkers": [ { "id": "11.8", "label": "Problem", "offset": 417 }, { "id": "11.8", "label": "Solution for Problem", "offset": 620 }, { "id": "1.3", "label": null, "offset": 1658 }, { "id": "0.70", "label": null, "offset": 1662 }, { "id": "0.65", "label": null, "offset": 1667 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0218.png", "sha256": "1d686acfec0bf2a9816290c01b192f39b589dd440814281f48e20366b7d07530", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 219, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ed9d94f8b7a88765b83a0498c253429428342496c709439f077d2184fffc1202", "textHash": "9381d52bf91c95aa65069c7743d7a05786ef5570ba067983d94326fcb7bd62dc", "characters": 1810, "flags": [], "candidateHeadings": [ "Important: All rational numbers have either terminating decimal expansions or repeating decimal expansions." ], "candidateMarkers": [ { "id": "11.9", "label": "Problem", "offset": 792 }, { "id": "11.9", "label": "Solution for Problem", "offset": 855 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0219.png", "sha256": "4157a0348de1c206f7b5c45e5003fd183660c95d4f0943415b6dc29aa3b919b8", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 220, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8a1ab09551aacb590ae72daadb87bbdf008e70c3ca72818a305651b26a8b5693", "textHash": "f89cd6d8b9d857ffa1b9179f6dd91dab03603e9346e036986224a557c647f6c3", "characters": 1592, "flags": [], "candidateHeadings": [ "11.4 Converting Decimals to Fractions" ], "candidateMarkers": [ { "id": "11.3.1", "label": null, "offset": 591 }, { "id": "11.3.2", "label": null, "offset": 920 }, { "id": "11.4", "label": null, "offset": 1019 }, { "id": "11.10", "label": "Problem", "offset": 1338 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0220.png", "sha256": "fa31b69c0c9d6566e29c09a08e62d0be2a4f378719de6a29dff422f63294051d", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 221, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "228f7f94f23be978b4f52e164c9a4ab832b1c6dd82fd2fd2da236b0814101573", "textHash": "34ca3a69c6a735ed8faf22edaf1f4aa99716e0c1c2d921c2369b47b4419d544e", "characters": 1650, "flags": [], "candidateHeadings": [ "Concept: Algebraic methods are useful in developing techniques to solve some arithmetic problems. In Problem 11.10 we used algebra to take advantage of the self-similarity of a repeating decimal. We assigned a variable to represent the repeating decimal and found a way to express the variable as a fraction, thereby giving us a way to convert a repeating decimal to a fraction." ], "candidateMarkers": [ { "id": "11.11", "label": "Problem", "offset": 35 }, { "id": "11.12", "label": "Problem", "offset": 285 }, { "id": "11.13", "label": "Problem", "offset": 412 }, { "id": "11.10", "label": "Problem", "offset": 469 }, { "id": "11.10", "label": "Solution for Problem", "offset": 534 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0221.png", "sha256": "d9c0c28c052fcb1b55fb4f933c72a997aeb1e01e2a1c7f60ce660efe7d043cf2", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 222, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "380349598a506bcb38b711d16b425c1ef5d0dfb31271d4e3b6efa72c29057aa6", "textHash": "7a82c593ec2c7cdebee30c389397481b927f18a8edc8faedd1cac97234e77679", "characters": 1565, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.11", "label": "Problem", "offset": 39 }, { "id": "11.11", "label": "Solution for Problem", "offset": 152 }, { "id": "11.12", "label": "Problem", "offset": 767 }, { "id": "11.12", "label": "Solution for Problem", "offset": 829 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0222.png", "sha256": "56433cdb276449460ef39f445ce1f1022790d8f770041720657d071b1b4ae996", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 223, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d4aaa0a8695104894b2f1b46926656379278a14eeadb221eddcf8a781517eaca", "textHash": "938edd1e31f141db058680824bd5c3cb7e36268f54f2661a5259471cfccf3494", "characters": 1671, "flags": [], "candidateHeadings": [ "0.9 = \\(\\frac{9}{101}\\) + \\(\\frac{9}{102}\\) + \\(\\frac{9}{103}\\) + \\(\\cdots\\) = \\(\\frac{9}{1-\\frac{1}{10}}\\) = 1." ], "candidateMarkers": [ { "id": "11.13", "label": "Problem", "offset": 35 }, { "id": "11.13", "label": "Solution for Problem", "offset": 92 }, { "id": "0.9", "label": null, "offset": 1331 }, { "id": "11.4.1", "label": null, "offset": 1520 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0223.png", "sha256": "f83de34083437de56f0ae6583042acec262e158886832f5abe8f3872d3bd76bd", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 224, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "95ca4a5dc9e20442be97b7c046bef02a4c2126a66a63ff76bcbb41adae3f4b9c", "textHash": "81d3312ed9c19bec7d447191bd3bf3183c7147c6b2152b9d96f7e36f1154b4bf", "characters": 1848, "flags": [], "candidateHeadings": [ "0.57 = 5.7-1 = \\(\\frac{57}{107}\\) = \\(\\frac{5}{7}\\) = 0.714285", "88.89 = 8.91 + 8.90 + 8.9-1 = \\(\\frac{889}{109}\\) = \\(\\frac{728}{9}\\) = 80.8", "Definition: In base number systems, the dot used to separate the integral part of a number from the fractional part is called the radix point. (Radix is just another name for the base of a number system.)" ], "candidateMarkers": [ { "id": "11.5", "label": null, "offset": 0 }, { "id": "0.57", "label": null, "offset": 313 }, { "id": "88.89", "label": null, "offset": 533 }, { "id": "11.14", "label": "Problem", "offset": 1010 }, { "id": "11.15", "label": "Problem", "offset": 1251 }, { "id": "11.16", "label": "Problem", "offset": 1526 }, { "id": "11.17", "label": "Problem", "offset": 1591 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0224.png", "sha256": "9377efc99d1ee74051bd81d2fb1dc06a41413524aeb686ff08b793d25599024d", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 225, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "51d7edbbca226dcf8df4789984cdfcf8da0bd87fcb4f431ae3479f798586aab8", "textHash": "6e763cc5e87db70b5f0ea4f6b50a927b6e43cb5f2c10fe8e243d9cf7690a1ac7", "characters": 1799, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.14", "label": "Problem", "offset": 35 }, { "id": "11.14", "label": "Solution for Problem", "offset": 88 }, { "id": "11.15", "label": "Problem", "offset": 933 }, { "id": "11.15", "label": "Solution for Problem", "offset": 1046 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0225.png", "sha256": "92169052a918aa00c56ece319c8acecd409f8cbd5005594700a90dac7697192b", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 226, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "82e71a787be019cc95f7cdae72f55a1d4d2a39e4b3036f58947623074b8a9cd7", "textHash": "89086509dbe5061aed88c088444a8962d3702305543c69d93980b1b1a046d2f0", "characters": 1740, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.5", "label": null, "offset": 0 }, { "id": "11.16", "label": "Problem", "offset": 44 }, { "id": "11.16", "label": "Solution for Problem", "offset": 109 }, { "id": "11.17", "label": "Problem", "offset": 397 }, { "id": "11.17", "label": "Solution for Problem", "offset": 458 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0226.png", "sha256": "0aa9bc5a7194834d814eb72ed66a8a80dd3820328376ec54fbd7a0aa9696cd66", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 227, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6849ba101346a44caf19d17fca986b014a5e8c02f17b3a2d8de66e0ce05f562a", "textHash": "8d1f7bb4572f9734338c01e2fcae354013e11c34b0869694e78e18535c39c189", "characters": 1440, "flags": [], "candidateHeadings": [ "11.6 Summary", "Definitions:", "Concept: Algebraic methods are useful in developing techniques to solve some arithmetic problems." ], "candidateMarkers": [ { "id": "11.5.1", "label": null, "offset": 35 }, { "id": "11.5.2", "label": null, "offset": 373 }, { "id": "11.6", "label": null, "offset": 480 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0227.png", "sha256": "01dc94face073eb1e75dae8aea0daa98c0f68bdf03eec546705e5f6f1531c953", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 228, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "689df0a8a61da359a99957879ca175c6e05af8204503b37bc113117365dd3c30", "textHash": "b468b0130b0695a6a75890827aee5e6424ff806f5d62784ce1af97fae55572bc", "characters": 1649, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Important: Facts about the relationships between fractions and decimals:", "Concept: When faced with a problem involving the properties of numbers or principles of arithmetic, consider whether it’s more illuminating to view the numbers in terms of decimals or fractions. The fact that either one of these forms might shed more light on a particular idea than the other is another example of how we use convenient mathematical forms to solve problems.", "REVIEW PROBLEMS", "11.18 Express each of the following fractions as decimal numerals.", "11.19 How many of the first 10 positive integers have reciprocals that are repeating decimals? (Source: MATHCOUNTS)" ], "candidateMarkers": [ { "id": "11.18", "label": null, "offset": 1128 }, { "id": "11.19", "label": null, "offset": 1528 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0228.png", "sha256": "46a999036e8efcebfd6aa5ae3c981e56c7a2e117ab9f2264fd5fcc3b582c0556", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 229, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "870cf2a0b2f416c1f278f8d99f50e3d4b5dda0bea9ae15ea93065c0323f51a20", "textHash": "3ebfc35806f9a8a57eb29c0d2478a64fa7305dc1265fb399c572d780d0fd6005", "characters": 1274, "flags": [], "candidateHeadings": [ "11.20 Compute: \\(\\frac{4! + 3!}{3! + 2!}\\). Express your answer as a decimal to the nearest hundredth.", "11.21 Find the \\(291^{st}\\) digit past the decimal point in the expansion of \\(\\frac{1}{37}\\).", "11.22 Express each of the following decimal numerals as a fraction in reduced form.", "Challenge Problems", "11.23 Convert the following fractions to the indicated number bases.", "11.24 Convert each of the following to base 10.", "11.25 Find the decimal equivalents for each of the following fractions.", "11.26 What is the 4037th digit following the decimal point in the expansion of \\(\\frac{1}{17}\\)? (Source: MATHCOUNTS) Hints: 152", "11.27 Evaluate the infinite geometric series," ], "candidateMarkers": [ { "id": "11.20", "label": null, "offset": 35 }, { "id": "11.21", "label": null, "offset": 160 }, { "id": "11.22", "label": null, "offset": 256 }, { "id": "11.23", "label": null, "offset": 478 }, { "id": "11.24", "label": null, "offset": 610 }, { "id": "11.25", "label": null, "offset": 691 }, { "id": "11.26", "label": null, "offset": 959 }, { "id": "11.27", "label": null, "offset": 1089 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0229.png", "sha256": "c85e52e25265cfcb54d5335b02b381f2c0b2d1186af5ab43284cb6b6a871055b", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 230, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c7f6c78680b94e37e9c255287b434d31395f7aadabeec1c559f24ebe7c53b135", "textHash": "751b36ec4a869c4195c33953b133c9ab73fa2ae15389353f5274868ae3e43be1", "characters": 1114, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "11.28 Let p be a prime number other than 2 or 5. What is the maximum possible number of digits in the repeating block of digits in 1/p? Hints: 70", "11.30 What are the first 18 digits in the decimal expansion of \\(\\frac{1}{998}\\)? Hints: 45, 89", "11.31 Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal \\(\\overline{ab}\\) is expressed as a fraction in lowest terms. How many different denominators are possible? (Source: AHSME) Hints: 5", "11.32 We noted that when \\(\\gcd(m, n) = 1\\), the fraction m/n has a terminating decimal representation if and only if the prime factorization of n contains no primes other than 2 or 5. Suppose we look at the expansion (past the radix point) of m/n in base 8—for what values of n will the radix expansion of m/n terminate and for what values of n will it repeat? Can you generalize to other number bases? Hints: 18" ], "candidateMarkers": [ { "id": "11.28", "label": null, "offset": 19 }, { "id": "11.29", "label": null, "offset": 166 }, { "id": "11.30", "label": null, "offset": 366 }, { "id": "11.31", "label": null, "offset": 463 }, { "id": "11.32", "label": null, "offset": 695 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0230.png", "sha256": "a76b2f810695a42f105305749a3aa0823e00f25898cd839581e07f4f2ef692a0", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 231, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c99c219f90423e2ae9305e0ea6e4b516fd4b84093cde7dd1c0674a1399215378", "textHash": "bc9cf063be3a75f8dc9659465e1ebc7a62529a404c2f3b4295d0652099b23350", "characters": 1788, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/page-0231.png", "sha256": "aa64f49073b5e1cf20612935a1b22f213b819cc52f2241344d6499aa317b2e65", "nextChapterHeadingPDF": 232 } }, { "source": "book-05", "pdfPage": 232, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d4db5a5a3cd020630128f1da2bba5447cedf9fe62c8ba8187bf5487bab55ed31", "textHash": "9f0a26bb94810f23d09b238c6e5868901a886e070d69fbff6330233ba79093d4", "characters": 1969, "flags": [], "candidateHeadings": [ "12.1 Introduction" ], "candidateMarkers": [ { "id": "12.1", "label": null, "offset": 231 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0232.png", "sha256": "498927cb0da360db4550608b4060cacd12cd444722cd925b46e49cb8d6256251", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 233, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c2a4635045d7a15ea1646c3e2131e6145f5152e60a5366850148804e86fb080c", "textHash": "3571bbe2d78f32d7fff6e26646adb3122c2353076c2f0aaf74f8278def4814fd", "characters": 1842, "flags": [], "candidateHeadings": [ "12.2 Congruence" ], "candidateMarkers": [ { "id": "12.2", "label": null, "offset": 47 }, { "id": "12.1", "label": "Problem", "offset": 74 }, { "id": "12.2", "label": "Problem", "offset": 929 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0233.png", "sha256": "99de2afb7da6906acfca516e78a932bd308e5ed1fa32d329428699eabfb05b51", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 234, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "56b62050ec357ec38f499112810b15406dd146a274067e017215a1f9a903e5d0", "textHash": "76ae2674b06bd48b0240fb42dcf0aec0563fcb00ce93170015ca190bdd7f851d", "characters": 7102, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.3", "label": "Problem", "offset": 17 }, { "id": "12.1", "label": "Problem", "offset": 736 }, { "id": "12.1", "label": "Solution for Problem", "offset": 1450 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0234.png", "sha256": "b8bedd6a0cf45587a8034e964debfdc5f50f55a3e1d722de310535f6b302a908", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 235, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e1dba3a27bcf3ff33ae210d2c7c7f93b0832b4efe699f412b9a3aab2e300a3ae", "textHash": "6bc85a0edddb304f6cb24ace2a625994c7de88663c3f8a08fb6ddf5cfb3b4522", "characters": 6615, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [ "Definition: A modulus is a system for counting using only the fixed set of integers \\(0, 1, 2, \\ldots, m - 1\\). When working in this modulus of \\(m\\) integers, we say that we are working with the integers \\(modu\\)." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0235.png", "sha256": "da861f9b42ff72f08415a205f04eef073d694cf46e3924506e0ea81e890f1b34", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 236, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7f0f89ab5f9a6898128c5177b0ed5c7df97e68932053bd2ae50dbda8dd1fa93b", "textHash": "7215734c1bc6668387b52f2b6a9626d01bd8d356d742601a9cc834936cdfef5f", "characters": 1954, "flags": [], "candidateHeadings": [ "Definition: We say that two integers are congruent or equivalent modulo m when their difference is a multiple of m. Otherwise, they are incongruent in modulo m.", "Important: For integers a and b, we say that \\(a \\equiv b \\pmod{m}\\) if and only if \\(\\frac{a - b}{m}\\) is an integer. Otherwise, \\(a \\neq b\\) (mod m)." ], "candidateMarkers": [ { "id": "12.2", "label": "Solution for Problem", "offset": 17 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0236.png", "sha256": "46d0702019742ef171e3a5c404302cf711995495b26eaeb8273eae04567b75fd", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 237, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a3459f095c24989f9e4fa66f0a5fb1841be1cfdd0b2e8009f52a9e728ea9f9a7", "textHash": "3d1b226d82d1aec21815d63a43416b44cc42845b4c2317f5812171d5dbb925fd", "characters": 2367, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.3", "label": "Problem", "offset": 47 }, { "id": "12.3", "label": "Solution for Problem", "offset": 191 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0237.png", "sha256": "f266b37e0095cf5b2852980cb3e2f747bdda239ad2f285238dc67205031a8f4c", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 238, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "b9f1cfdbd3b5aa3884572e0e8f8f0b9c59ce4a763f87008e6d4fdbdcd8d235d7", "textHash": "a90d020f999379cef9d26a31040fb3e4a49f1d0b13e59660eb3f4d428d41a30b", "characters": 1581, "flags": [], "candidateHeadings": [ "Important: Parametric expressions give us another way to tell when two integers are congruent in a given modulus:", "Exercises" ], "candidateMarkers": [ { "id": "12.2.1", "label": null, "offset": 397 }, { "id": "12.2.2", "label": null, "offset": 1321 }, { "id": "12.2.3", "label": null, "offset": 1459 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0238.png", "sha256": "63646f779e7b88ce1846b0fdc8000ecc7b64bfe61d358e49cc97f897de2a55ac", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 239, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "bf4b74f49828e6571b0d83eab8bc245d9be9ff5f9b29130f2bc29106b940992c", "textHash": "6b752dd563acd219706e7cc125686a3c0d9a6a7fe32b43e8d5863d508bae46a8", "characters": 1539, "flags": [], "candidateHeadings": [ "12.3 Residues" ], "candidateMarkers": [ { "id": "12.2.4", "label": null, "offset": 47 }, { "id": "12.2.5", "label": null, "offset": 165 }, { "id": "12.2.6", "label": null, "offset": 399 }, { "id": "12.3", "label": null, "offset": 480 }, { "id": "12.4", "label": "Problem", "offset": 505 }, { "id": "12.5", "label": "Problem", "offset": 907 }, { "id": "12.6", "label": "Problem", "offset": 994 }, { "id": "12.4", "label": "Problem", "offset": 1215 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0239.png", "sha256": "48ce3e6b65aadfbd5b636b5f44f2c24aa9f0df39d944e0a9d8a4cf994386052b", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 240, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d02f89e6c2f5ff773ecc94dcb84dcd623cb46ea171ceb312c43346a4bd03ef35", "textHash": "e22bc345f33a639b1a63afe35a75fb3edbd937a9ede652bbc3a64de8f717f813", "characters": 2071, "flags": [], "candidateHeadings": [ "Definition: We say that \\( r \\) is the modulo \\( m \\) residue of \\( n \\) when \\( n \\equiv r \\pmod{m} \\) and \\( 0 \\leq r < m \\)." ], "candidateMarkers": [ { "id": "12.4", "label": "Solution for Problem", "offset": 15 }, { "id": "12.5", "label": "Problem", "offset": 836 }, { "id": "12.5", "label": "Solution for Problem", "offset": 937 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0240.png", "sha256": "97a12d7dcfc54f9de03c6e9dcae9f96ef98c8749f7d91c9a8b86a0f0b630686a", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 241, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fc4847a8db2686d2f2755d6079c114794c97e06d019a5834e033ff0419b6385f", "textHash": "b84ec36033827cc94558a0afe7a05dae2f2060148819fc385d244575df7cd039", "characters": 2326, "flags": [], "candidateHeadings": [ "Definition: The integers congruent to \\( r \\) modulo \\( m \\) make up a residue class. Residue classes are also known as congruence classes or equivalence classes.", "Important: If \\( a \\equiv b \\pmod{m} \\) and \\( b \\equiv c \\pmod{m} \\), then \\( a \\equiv c \\pmod{m} \\).", "Exercises" ], "candidateMarkers": [ { "id": "12.6", "label": "Problem", "offset": 604 }, { "id": "12.6", "label": "Solution for Problem", "offset": 736 }, { "id": "12.3.1", "label": null, "offset": 1921 }, { "id": "12.3.2", "label": null, "offset": 1961 }, { "id": "12.3.3", "label": null, "offset": 2141 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0241.png", "sha256": "0030dad6d12c3839105ee62bd2043e0702e30eb53a6035b7ba396c10066e5bd6", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 242, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1651d2d6fe31cfbf0edc5f42b783ee4f19f8a78f21560d9cba086a2a51b7eaf7", "textHash": "c8f79dbc033a71801a95b81bf56344f32fc8dbb849031ba32181f9906cae045f", "characters": 1163, "flags": [], "candidateHeadings": [ "12.4 Addition and Subtraction" ], "candidateMarkers": [ { "id": "12.3.4", "label": null, "offset": 31 }, { "id": "12.3.5", "label": null, "offset": 159 }, { "id": "12.4", "label": null, "offset": 318 }, { "id": "12.7", "label": "Problem", "offset": 551 }, { "id": "12.8", "label": "Problem", "offset": 566 }, { "id": "12.8", "label": "Problem", "offset": 837 }, { "id": "12.9", "label": "Problem", "offset": 1104 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0242.png", "sha256": "d34dd70c4e10f3ae6fa92a58e9fab95af5096358fd8ccd2617453776cdbd09be", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 243, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8e981a27d1ccd281b7b29fdd94f08370ed14fae952f04eb04edcc8cd8a89f049", "textHash": "4cef8ca0fa00357165d0713ccc9862e3c22fd302def4892d47277ee1c2ff6a9a", "characters": 2284, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.9", "label": "Problem", "offset": 47 }, { "id": "12.10", "label": "Problem", "offset": 236 }, { "id": "12.11", "label": "Problem", "offset": 450 }, { "id": "12.7", "label": "Problem", "offset": 862 }, { "id": "12.7", "label": "Solution for Problem", "offset": 1128 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0243.png", "sha256": "e64617631683deb7193e8842e86191352ee369cdc59ff2589917b158dc3548ed", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 244, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e7f39ddc321e271bf96ec912fdad0b4e9b9dcb20d4b1cdf29839a943c2d9be25", "textHash": "4043a652fa409ca2ac06dfaff047453175e91f45c40a346b2552ca7e7b421d29", "characters": 1948, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.8", "label": "Problem", "offset": 1490 }, { "id": "12.8", "label": "Solution for Problem", "offset": 1670 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0244.png", "sha256": "a69bb9d420a8e0dc622aa3e3d2bbe02c6c0437045804bfff2cfc603224d6f070", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 245, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f793519bdf9fd6e665321fff0255eaa1e7c7bec52553e5b15e7e945a7e1ad859", "textHash": "2738f7de2e57e1aa980e7c7381f9c1b73266a4ee4b2c8e004792049e2d079c50", "characters": 1643, "flags": [], "candidateHeadings": [ "Important: Let \\(a_1, a_2, b_1\\), and \\(b_2\\) satisfy" ], "candidateMarkers": [ { "id": "12.9", "label": "Problem", "offset": 1073 }, { "id": "12.9", "label": "Solution for Problem", "offset": 1276 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0245.png", "sha256": "477198efc60136407a3d45414fd25fcb92c61f9ab9d496fa30537e6f612567df", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 246, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7607c2b96044d597afa5412bbcf17a0bf7b6f149c1471c1b08a462cb101c941e", "textHash": "76422fe7600dec35fed604c8c2bed5c2bba64fe34d43682de721097eddede8c1", "characters": 1774, "flags": [], "candidateHeadings": [ "Concept: Modular arithmetic helps us solve problems about unknown integers by giving us a way to work with their incomplete descriptions." ], "candidateMarkers": [ { "id": "12.10", "label": "Problem", "offset": 105 }, { "id": "12.10", "label": "Solution for Problem", "offset": 320 }, { "id": "12.11", "label": "Problem", "offset": 1134 }, { "id": "12.11", "label": "Solution for Problem", "offset": 1537 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0246.png", "sha256": "9cd1e4ca37e3bcde8925cbfd2a21d9271f29390ea22dd9a62070b22a2016e045", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 247, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "79e3f83566b66288371d04bc50f661648850ffd8a59183d3ae9ac4e6ae2878a8", "textHash": "1b259c97a34d93e9cad39d9a6679bc19b8565e3afef6ec785e58f7eec81ce06c", "characters": 2062, "flags": [], "candidateHeadings": [ "Exercises", "12.5 Multiplication and Exponentiation" ], "candidateMarkers": [ { "id": "12.4.1", "label": null, "offset": 163 }, { "id": "12.4.2", "label": null, "offset": 501 }, { "id": "12.4.3", "label": null, "offset": 618 }, { "id": "12.4.4", "label": null, "offset": 927 }, { "id": "12.4.5", "label": null, "offset": 1075 }, { "id": "12.5", "label": null, "offset": 1442 }, { "id": "12.12", "label": "Problem", "offset": 1492 }, { "id": "12.13", "label": "Problem", "offset": 1756 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0247.png", "sha256": "9275fb473d4ec21492715eced257ba39e21fcbaee43c8c7ed4d676469fffaecc", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 248, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4097ef612085e17e3d93cef15845864c58a898cd16ef92262444f9308aa5f188", "textHash": "8638395b72d9588eda1224bc8fb8f1178830a031a1558d792cfed38db7ae720f", "characters": 1586, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.14", "label": "Problem", "offset": 40 }, { "id": "12.15", "label": "Problem", "offset": 246 }, { "id": "12.16", "label": "Problem", "offset": 535 }, { "id": "12.17", "label": "Problem", "offset": 1079 }, { "id": "12.18", "label": "Problem", "offset": 1252 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0248.png", "sha256": "27636cccbeaca5bec9907b581aea1941c26fba82929d83e644d4d1a64cf3b5b8", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 249, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "53e1edf46e3f35aa5aec164b8f4de61bdac163db84d1dea1926cdd351a72fad5", "textHash": "20171d076a199b5fcbeae0c9bdf5c3f0ee96942b36586e2a062ff25a435bc86d", "characters": 1716, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.12", "label": "Problem", "offset": 47 }, { "id": "12.12", "label": "Solution for Problem", "offset": 205 }, { "id": "12.13", "label": "Problem", "offset": 710 }, { "id": "12.13", "label": "Solution for Problem", "offset": 895 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0249.png", "sha256": "52c8c8e596a7b6c49409354d57c81c71efe70c1c2f34fa3fcd68f36d3646ae91", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 250, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9e2725e63c878507af002a022293b979df84009278f7a66d6a96e20d9edafe44", "textHash": "8d831faea2b28beea9d647c147ed783829baec09817fa197b49417278d2738b1", "characters": 1579, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.14", "label": "Solution for Problem", "offset": 891 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0250.png", "sha256": "22378d68d454363d7a4ac8276d8959376cbd2425e8ed1029c9d7a2a00d196d79", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 251, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4b3c74d1c93ca6c18a50aaa07bdc0c90388a49c557aa76079475c9755350fbe7", "textHash": "cf42763f8765b1d3828dc3ab62c1a4f82adca7fa28076e7e079d2549b9df6e03", "characters": 1653, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.15", "label": "Problem", "offset": 47 }, { "id": "12.15", "label": "Solution for Problem", "offset": 281 }, { "id": "12.16", "label": "Problem", "offset": 582 }, { "id": "12.16", "label": "Solution for Problem", "offset": 1126 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0251.png", "sha256": "28a6fb4f11cd18330958ae1d9fa16d51ba11c55dfd6799caa9c39d5e92bd872c", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 252, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "15b049d425df6a54e38981bd30362749ae7c8cb34f6bf11f03b6fa551d9dfed8", "textHash": "525a374458f9d0500750368894c118923b6a0036600b51457d9568b70312dd04", "characters": 2467, "flags": [], "candidateHeadings": [ "Concept: Modular arithmetic becomes a powerful tool when combined with an understanding of algebra.", "Concept: In the solution to Problem 12.17, we used a process called induction to build on a simple result to establish a more general result. The idea is that each step along the way is essentially the same. Each new congruence is a direct result of the previous congruence and the process continues endlessly.", "Important: For any natural number \\( n \\) and integers \\( a_1 \\) and \\( a_2 \\), where \\( a_1 \\equiv a_2 \\) (mod \\( m) \\), \\( a_1^2 \\equiv a_2^2 \\) (mod \\( m) \\)." ], "candidateMarkers": [ { "id": "12.17", "label": "Problem", "offset": 141 }, { "id": "12.17", "label": "Solution for Problem", "offset": 279 }, { "id": "12.18", "label": "Problem", "offset": 1638 }, { "id": "12.18", "label": "Solution for Problem", "offset": 1701 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0252.png", "sha256": "ebfc1746754b561027d9f92488c3542cfb02227c2104fde43e4f4961980dc70b", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 253, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "95744ab3b65886bb877bd2784dfd0019cb11df359675dd92fe8fcc583b256a02", "textHash": "2a1b2fbf0723fc8325a50e2bdb0550558a00d40d3f2f556608882348007e9761", "characters": 1563, "flags": [], "candidateHeadings": [ "Concept: In modular arithmetic, we usually work with residues because they typically make the arithmetic easiest. However, negative integers are easier to work with in some cases. It's always good to keep an open mind to the possibility of simpler solutions.", "Exercises", "12.6 Patterns and Exploration" ], "candidateMarkers": [ { "id": "12.5.1", "label": null, "offset": 318 }, { "id": "12.5.2", "label": null, "offset": 466 }, { "id": "12.5.3", "label": null, "offset": 526 }, { "id": "12.5.4", "label": null, "offset": 667 }, { "id": "12.5.5", "label": null, "offset": 726 }, { "id": "12.5.6", "label": null, "offset": 1053 }, { "id": "12.6", "label": null, "offset": 1115 }, { "id": "12.19", "label": "Problem", "offset": 1156 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0253.png", "sha256": "47bbccdb5e13590a3c483ba30a667975bb119e9a17dad929fc2408361de89f21", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 254, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f825f96e11d499745f3042935febc6ac78ebcbd18eb9f622ad231119933d6de7", "textHash": "e1c59bc4c99580a6f7daa2a5df72eae8e32749dd75cbc4d346b67bd62893b2a6", "characters": 2391, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.20", "label": "Problem", "offset": 31 }, { "id": "12.21", "label": "Problem", "offset": 201 }, { "id": "12.22", "label": "Problem", "offset": 367 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0254.png", "sha256": "bd4f4ab641cd84312579ecd24560ff05b6d8b87af0fef0d256d8739e6b71be0a", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 255, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6cea18433e8a36f68bca2828e4c9ec6ba322f8071756588d72cefc6663b31c1d", "textHash": "511d5af73ea28cc7e99c753ed8bb97430155f5f6939b8982246c24798713ba90", "characters": 1738, "flags": [], "candidateHeadings": [ "Concept: When faced with a problem that requires a difficult or impossible calculation, hunting for patterns often helps. This is especially true in modular arithmetic problems. The limited number of residues within a modulus sometimes make it easy to spot useful patterns." ], "candidateMarkers": [ { "id": "12.19", "label": "Problem", "offset": 47 }, { "id": "12.19", "label": "Solution for Problem", "offset": 119 }, { "id": "12.20", "label": "Problem", "offset": 1352 }, { "id": "12.20", "label": "Solution for Problem", "offset": 1436 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0255.png", "sha256": "073a2a85b26fddbbe7ed7b6d4646b3a2fe1e5b1f1e10266501e71706dcf42872", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 256, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6ebc5baaf3cbb6578947ba75e34437f9d5a4b9b83e9d6985a16b4341d5b5a5eb", "textHash": "6ea3c8efb7a840eb84ccac52fc5bacfd39937173cd120cf9385c988bf0136bd2", "characters": 1701, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "12.21", "label": "Problem", "offset": 31 }, { "id": "12.21", "label": "Solution for Problem", "offset": 197 }, { "id": "12.22", "label": "Problem", "offset": 587 }, { "id": "12.22", "label": "Solution for Problem", "offset": 636 }, { "id": "12.6.1", "label": null, "offset": 1365 }, { "id": "12.6.2", "label": null, "offset": 1497 }, { "id": "12.6.3", "label": null, "offset": 1656 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0256.png", "sha256": "0c3671eddd1b73d55211a524a7a39e237e7e4591cf5fce1dd6a939331d92d239", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 257, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "76ce04005bc720cd357a5ee692119f2df7cb0e8f73cd59fd41f831ab5f074da9", "textHash": "abf322ea58df6c8a16fd704213a49934116dad7d3c3821e2aaef40cb931d9fa8", "characters": 1786, "flags": [], "candidateHeadings": [ "12.7 Summary", "Definitions:", "Concept: Congruence works in modular arithmetic much the same way that equality works with regular integers (or any numbers in the regular system of real numbers). However, instead of meaning that two integers are the same, congruence means that the integers occupy the same column in a counting grid in which the number of columns is the modulus of the system." ], "candidateMarkers": [ { "id": "12.6.4", "label": null, "offset": 47 }, { "id": "12.6.5", "label": null, "offset": 102 }, { "id": "12.6.6", "label": null, "offset": 188 }, { "id": "12.7", "label": null, "offset": 290 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0257.png", "sha256": "32217778ff8edf33f495440481c284a299dd886a097516ea83ebdf48a7f9331d", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 258, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fc910fcb2667c5b2320a2b7ddaa0dc91d37761b38f310c480375c5df89471e71", "textHash": "b5b909b9a3dfac913c1bc3778b5c81bef8ad7157203403265ce00a6e733b72d7", "characters": 1456, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Important: Let \\(a_1\\), \\(a_2\\), \\(b_1\\), and \\(b_2\\) be integers such that", "Concept: Part of what makes modular arithmetic useful is that we can usually simplify calculations by working with the residues of integers. However, be aware of exceptions. It's often easiest to work with \\(-1\\) or other integers with convenient properties relating to a particular problem.", "Concept: When faced with a problem that requires a difficult or impossible calculation, hunting for patterns often helps. This is especially true in modular arithmetic problems. The limited number of residues within a modulus sometimes make it easy to spot useful patterns.", "REVIEW PROBLEMS", "12.23 Find the modulo 6 residues of each of the following.", "12.24 Which of the following statements of congruence are true and which are false?" ], "candidateMarkers": [ { "id": "12.23", "label": null, "offset": 952 }, { "id": "12.24", "label": null, "offset": 1157 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0258.png", "sha256": "051f82f19eb27e42220a6f54839794fa411bf63ecc1cf263bef325851c1746d4", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 259, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "db51eddca323aa1166060e45b32bdc91bcc8a240fa4c59b970b3c1e88e72235d", "textHash": "53fb84c5fbf80dc989aec46947b6ce9a48226d17e247ed1387bd13df10771281", "characters": 7015, "flags": [ "truncated" ], "candidateHeadings": [ "12.25 How many integers between 1000 and 3000 inclusive are congruent to 5 (mod 7)?", "12.26 For how many values of n, where 40 ≤ n ≤ 80 is n ≡ -n (mod 12)?", "12.27 Four jars of jelly beans are individually repackaged into packs of 12 jelly beans. After the maximum number of packs are filled from each of these jars, the numbers of remaining jelly beans in the jars are 3, 5, 7, and 11. If the four jars had been combined before repackaging and the maximum number of packs were filled, how many jelly beans would have been left out of the 12 packs?", "12.28 Two positive integers leave remainders of 6 and 13 when divided by 14.", "12.29 Prove that if a ≡ 19 (mod 30), then 3a ≡ 7 (mod 10).", "12.30 Find the remainder when 7255 is divided by 11.", "12.31 Find the remainder when 942 − 542 is divided by 7.", "12.32 John tells Susie that he’s thinking of an integer that leaves a remainder of 2 when divided by 6. Susie correctly informs John that the number is not a perfect square. How does she know?", "Challenge Problems", "12.34 Find the modulo 200 residue of the sum", "12.35 Find the smallest natural number n such that", "12.36 Find the remainder when" ], "candidateMarkers": [ { "id": "12.25", "label": null, "offset": 47 }, { "id": "12.26", "label": null, "offset": 132 }, { "id": "12.27", "label": null, "offset": 203 }, { "id": "12.28", "label": null, "offset": 595 }, { "id": "12.29", "label": null, "offset": 915 }, { "id": "12.30", "label": null, "offset": 975 }, { "id": "12.31", "label": null, "offset": 1029 }, { "id": "12.32", "label": null, "offset": 1087 }, { "id": "12.33", "label": null, "offset": 1301 }, { "id": "12.34", "label": null, "offset": 1409 }, { "id": "12.35", "label": null, "offset": 1499 }, { "id": "12.36", "label": null, "offset": 1585 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0259.png", "sha256": "db1b6710097ed2e96dda40fb135e560e2050b0faf0018e278a46d7e29f11b661", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 260, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fc2cbe4f0264499048cc479f571be1940959f5ac2fcba92dc9086bb1605f79c4", "textHash": "7a27588c7f1eb2e91c8ce5282233b8e63694a5d903e322406792910956d14d6c", "characters": 2668, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "12.37 Suppose a and b are positive integers, neither of which is a multiple of 3. Find all possible remainders when \\(a^2 + b^2\\) is divided by 3. (Source: UNCC) Hints: 2", "12.38 Ryun and Zhenya play a game with a pile of 82 toothpicks. The players take turns removing 1, 2, 3, or 4 toothpicks from the pile on each turn. The player that removes the last toothpick loses. Zhenya goes first. Help her formulate a winning strategy. Hints: 37, 71", "12.39 A natural number \\(n\\), has a units digit of A when expressed in base 12. Find the remainder when \\(n^2\\) is divided by 6. Hints: 124", "12.40 For each integer \\(N > 1\\), there is a mathematical system in which two or more integers are defined to be congruent if they leave the same non-negative remainder when divided by N. If 69, 90, and 125 are congruent in one such system, then what is the residue of 81 in that system? (Source: AMC) Hints: 77", "12.41 A certain natural number is congruent to 4 (mod 9), congruent to 1 (mod 5), and congruent to 5 (mod 8).", "12.42 Find the remainder when", "12.43 A group of 25 Chicago Cubs fans got together for a party to discuss how this is the year. At the end of the party, many of the Cubs fans shake hands with one another. Let \\(n\\) be the number of the Cubs fans who shook hands with an odd number of other Cubs fans.", "12.44 What is the size of the largest subset, \\(S\\), of \\(\\{1, 2, 3, \\ldots, 50\\}\\) such that no pair of distinct elements of \\(S\\) has a sum divisible by \\(7\\)? (Source: AMC) Hints: 81, 50", "12.45 The Fibonacci sequence is defined by \\(F_1 = F_2 = 1\\) and \\(F_{n+2} = F_{n+1} + F_n\\). Find the remainder when \\(F_{2006}\\) is divided by 5. Hints: 147", "12.46 Find all prime numbers \\(p\\) for which \\(p^2 - 1\\) is not a multiple of 24. Hints: 105", "12.47 Prove that among any 51 integers, there are two whose squares have the same remainder when divided by 100. Hints: 151", "12.48 Denote by \\(p_k\\) the \\(k^{th}\\) prime number. Show that, for any positive integer \\(n\\), the number \\(p_1 p_2 p_3 \\cdots p_n + 1\\) cannot be the perfect square of an integer. (Source: M&IQ 1992) Hints: 88, 155" ], "candidateMarkers": [ { "id": "12.37", "label": null, "offset": 19 }, { "id": "12.38", "label": null, "offset": 191 }, { "id": "12.39", "label": null, "offset": 463 }, { "id": "12.40", "label": null, "offset": 604 }, { "id": "12.41", "label": null, "offset": 917 }, { "id": "12.42", "label": null, "offset": 1220 }, { "id": "12.43", "label": null, "offset": 1374 }, { "id": "12.44", "label": null, "offset": 1771 }, { "id": "12.45", "label": null, "offset": 1962 }, { "id": "12.46", "label": null, "offset": 2122 }, { "id": "12.47", "label": null, "offset": 2216 }, { "id": "12.48", "label": null, "offset": 2341 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0260.png", "sha256": "43d7f405b4abe875bcf27ca210ea3fad76a817a540bd666e046783abaf5b9261", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 261, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6af47dd19fb11cb1aec2f135d855df6176ce64522e867efd8b3ffa9a66ee985a", "textHash": "d2faf57ded405487e9a969982709483d0c7271af0246b8740cd59a2c58b5fc3c", "characters": 6035, "flags": [ "truncated" ], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/page-0261.png", "sha256": "55db73de59eff6fdd6d96a807c907f802c121333a8e596ec8336a25ae5886590", "nextChapterHeadingPDF": 262 } }, { "source": "book-05", "pdfPage": 262, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e96cbcfe69e6dae23cec54ea98ca215f1078806b8abbbc7be11d364718b73d93", "textHash": "b1b27b4180cca7b9925a442a635afa15fb7212150d1530e1f3e021c16cf54497", "characters": 1209, "flags": [], "candidateHeadings": [ "13.1 Introduction", "13.2 Divisibility Rules" ], "candidateMarkers": [ { "id": "13.1", "label": null, "offset": 152 }, { "id": "13.2", "label": null, "offset": 829 }, { "id": "13.1", "label": "Problem", "offset": 984 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0262.png", "sha256": "31a19b064917ef884fa2fea06ff10f1e70f82504113ce14a25b263c55ef945bf", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 263, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9e7cd8c9d4971e76734771441dfe116f5ae1f80c96c630ef71ecec48451b4348", "textHash": "f8d49f01623da7c0ee0575fd86885e656819dd4280cf80c4f4b39436a7210df7", "characters": 1871, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "13.2", "label": "Problem", "offset": 31 }, { "id": "13.3", "label": "Problem", "offset": 515 }, { "id": "13.4", "label": "Problem", "offset": 814 }, { "id": "13.5", "label": "Problem", "offset": 1154 }, { "id": "13.6", "label": "Problem", "offset": 1494 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0263.png", "sha256": "d4fe02bc6fb517e9a568ffca4e7e796d4118bcd8bc35136272b610060e97849d", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 264, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "83be4341f8f85aad7cf71fc044fef8c6f705807489dc369ed6b21f902ff4fcfe", "textHash": "f9845048bdaa2ac27adee579d19b876203dc3fffc9842a66a98b986b56344d2b", "characters": 2121, "flags": [], "candidateHeadings": [ "Important:" ], "candidateMarkers": [ { "id": "13.7", "label": "Problem", "offset": 25 }, { "id": "13.1", "label": "Problem", "offset": 206 }, { "id": "13.1", "label": "Solution for Problem", "offset": 335 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0264.png", "sha256": "8bdc0e54a2d1acbb10697f01e4693056529e883c34f4e42c6d23c0632b6e4b6e", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 265, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "448bec800f5be5eb7bd0ffac35346b81599532da54a640aeb2a11df0e45ea759", "textHash": "e9201ba000cbac309229742178947a5c3d575125d933810ba461aecb3a74a4c5", "characters": 2312, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "13.2", "label": "Problem", "offset": 31 }, { "id": "13.2", "label": "Solution for Problem", "offset": 180 }, { "id": "13.3", "label": "Problem", "offset": 925 }, { "id": "13.3", "label": "Solution for Problem", "offset": 1188 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0265.png", "sha256": "caf6d31f8c7162c1a7819dba727569cfa5a1787de956405c64f43fae4a07c6d8", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 266, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ecd90920ca6b1d94ad0e87c5ad0484191a9083a87175d512eaab7be2494af2de", "textHash": "c32034243473241b030e884958b9fd5bbcdfd1ace947f04f897958c62daa91de", "characters": 2357, "flags": [], "candidateHeadings": [ "Important: Let \\( m \\) be a positive integer." ], "candidateMarkers": [ { "id": "13.4", "label": "Problem", "offset": 790 }, { "id": "13.4", "label": "Solution for Problem", "offset": 880 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0266.png", "sha256": "cd201f523ba3ad167b13f19a5e641e37f81580886d74febfde46ab7818902288", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 267, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c837085f7393ec2f92747d1c397e373b9da25696c347bd22b2820241f9fe2370", "textHash": "f28b892afdf1f93b023a03e6248427f2b0f3a61f614f382c84fa848cd412d074", "characters": 2349, "flags": [], "candidateHeadings": [ "Important: An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.", "Important: An integer is divisible by 9 if and only if the sum of its digits is divisible by 9." ], "candidateMarkers": [ { "id": "13.5", "label": "Problem", "offset": 388 }, { "id": "13.5", "label": "Solution for Problem", "offset": 478 }, { "id": "13.6", "label": "Problem", "offset": 2063 }, { "id": "13.6", "label": "Solution for Problem", "offset": 2160 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0267.png", "sha256": "cc76274ea37f3bbab0c835e9481d13ac268f9ffb40678b7aced7c98acbe2ee3e", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 268, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d332dec2e0e8cddc4aaeb230f31af6fb58d57b75f45963beeaa7faa363620649", "textHash": "10732fb4afb9597cd17630eb142bc7b6b40cd3b050784a135b4b75c4270d9a68", "characters": 2553, "flags": [], "candidateHeadings": [ "Important: An integer is a multiple of 11 if and only if the alternating sum of its digits is a multiple of 11." ], "candidateMarkers": [ { "id": "13.7", "label": "Problem", "offset": 1948 }, { "id": "13.7", "label": "Solution for Problem", "offset": 2129 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0268.png", "sha256": "35732c359a3a1c5f19203a992bf0e42c3999483c8b8123a352a214e3768796ea", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 269, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "f0ef9e830820999285dcf6845da6a1c2e9e49cb0dc29f4921c298f13cf2bbbc3", "textHash": "1bc3e49d7ae8fd69831c569dd1faa6114da029d82fec9ee6a6933d43c4a44df4", "characters": 2623, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0269.png", "sha256": "22bc4e5859c651e28212cfa8e26b1aa93826c8465a22f6813281b7175de6be70", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 270, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e0a755349579ea33cb4e179bac31de6c1c44efbda052d12cf013111bfb4d321e", "textHash": "2dc678a0e6bcc801d444a1ed14cffa23b2f45925640cade0febe4176fdb0dade", "characters": 1558, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "13.3", "label": null, "offset": 0 }, { "id": "13.2.1", "label": null, "offset": 50 }, { "id": "13.2.2", "label": null, "offset": 210 }, { "id": "13.2.3", "label": null, "offset": 344 }, { "id": "13.2.4", "label": null, "offset": 478 }, { "id": "13.3", "label": null, "offset": 575 }, { "id": "13.8", "label": "Problem", "offset": 974 }, { "id": "13.9", "label": "Problem", "offset": 1075 }, { "id": "13.10", "label": "Problem", "offset": 1385 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0270.png", "sha256": "be43d54e073a248e82dbf35910a7017270678cc3c2c14c6fbb994f2541a048f0", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 271, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6832747fe4d7f7184f671253c151924528031f4d6cd8aa676dbfd50bf76e4d61", "textHash": "0bd14b2d02d1f9aab1c57b718fa4c9ea4955c25d15a6e1d0da71edfc6cb57ac1", "characters": 2180, "flags": [], "candidateHeadings": [ "Concept: When number theory problems involve variables, we can examine the information in the problem using the same techniques we apply to simpler problems. Then we apply algebraic concepts to solve those problems." ], "candidateMarkers": [ { "id": "13.8", "label": "Problem", "offset": 31 }, { "id": "13.8", "label": "Solution for Problem", "offset": 91 }, { "id": "13.9", "label": "Problem", "offset": 630 }, { "id": "13.9", "label": "Solution for Problem", "offset": 732 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0271.png", "sha256": "87844e61c7b010a031573a7978f7ac179eb373d85796eb2027570ad98cd3cbaf", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 272, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "e2bea960f32e38ca030f907771500dcb5eca8666f3e986f12b18b9059f9c0370", "textHash": "535e0127ed315f0cd4b4e2067702875f12d698ec11ef04084ecaf0902f3f4668", "characters": 2005, "flags": [], "candidateHeadings": [ "Concept: While there might be no simple rule for how to solve a particular problem, we can often use several rules together to complete the task.", "Exercises" ], "candidateMarkers": [ { "id": "13.3", "label": null, "offset": 0 }, { "id": "13.10", "label": "Problem", "offset": 469 }, { "id": "13.10", "label": "Solution for Problem", "offset": 562 }, { "id": "13.3.1", "label": null, "offset": 1272 }, { "id": "13.3.2", "label": null, "offset": 1326 }, { "id": "13.3.3", "label": null, "offset": 1415 }, { "id": "13.3.4", "label": null, "offset": 1504 }, { "id": "13.3.5", "label": null, "offset": 1606 }, { "id": "13.3.6", "label": null, "offset": 1828 }, { "id": "13.3.7", "label": null, "offset": 1918 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0272.png", "sha256": "4bf179fe82f3872511127ac2cc188c524574182239695eade71368b5e7f4f90a", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 273, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fb9c48bcf6a35bb4a503cdb1476f1a16177865ba7d2691b82fa756a99219a312", "textHash": "107383192b400643c98acd84f62058c2a21ce42b486f78a377161144cf2d4c3e", "characters": 1909, "flags": [], "candidateHeadings": [ "13.4 Summary", "Important: We used modular arithmetic to demonstrate relationships between the values of integers and their digits.", "Important: Here is a list of the most commonly used divisibility rules. When used, m represents any natural number." ], "candidateMarkers": [ { "id": "13.4", "label": null, "offset": 31 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0273.png", "sha256": "85dfef71a9987c97e281f7dddd6f67082a85f5a3209ccc56bd5bcf90b1ca4b59", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 274, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c3fb9f1440ec39f362847c4f0ebd3a643d9d6556a800f11ffe62853c17dd1868", "textHash": "03c90c8c74f5c9f50aa1708268cbe6fbd354f1e4d4beedccf38ed16ba170d95a", "characters": 1736, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Concept: When number theory problems involve variables, we can examine the information in the problem using the same techniques we apply to simpler problems. Then we apply algebraic concepts to solve those problems.", "Concept: While there might be no simple rule for how to solve a particular problem, we can often use several rules together to complete the task.", "REVIEW PROBLEMS", "13.11 Determine which of the following are multiples of 9.", "13.12 Determine which of the following are multiples of 11.", "13.13 Determine which of the following are multiples of 8.", "13.14 If A and B represent digits of the base ten number 60A5B, which is a multiple of 72, what are A and B? Hints: 112", "13.15 If A and B represent digits of the base ten number 50A11B, which is a multiple of 45, what are the possible values of 50A11B? Hints: 44" ], "candidateMarkers": [ { "id": "13.11", "label": null, "offset": 570 }, { "id": "13.12", "label": null, "offset": 671 }, { "id": "13.13", "label": null, "offset": 771 }, { "id": "13.14", "label": null, "offset": 876 }, { "id": "13.15", "label": null, "offset": 997 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0274.png", "sha256": "8eec780f49937977f82323f8df1aa1af41a9d9cc91021387d0c4b35ace62003e", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 275, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "438457cc1c7795b112565c8666a4de3b77797e059df3813d898ce6bc27f09193", "textHash": "216a2dc9168721ca9abcb40f9dee4614374f076916523c04c89bf8efab2e1f4d", "characters": 2440, "flags": [], "candidateHeadings": [ "Challenge Problems", "13.16 A nine-digit number uses the digits 1-9 once each. What is the probability that the number is prime? (Source: MATHCOUNTS) Hints: 119", "13.17 A four-digit number uses each of the digits 1, 2, 3 and 4 exactly once. What is the probability that the number is a multiple of 4? (Source: MATHCOUNTS) Hints: 63", "13.18 How many four-digit palindromes are multiples of 9? Hints: 36", "13.20 To test a number for divisibility by 7, remove the final digit, then subtract twice this digit from whatever remains. The new number is divisible by 7 if and only if the original one is. For example, 7 divides 434 since it divides 43 – 2(4) = 35. If instead we subtract three times the final digit from what remains, we obtain a divisibility test for a different prime. What is it? (Source: Mandelbrot) Hints: 84", "13.21 How many three-digit natural numbers have digits that sum to 13 and leave a remainder of 1 when divided by 4? Hints: 109", "13.22 Find a six-digit number whose first three digits are 523 such that the integer is divisible by each of 7, 8, and 9. Hints: 11", "13.24 Show that a natural number \\( n \\) is a multiple of \\( b - 1 \\) if and only if the sum of its digits are a multiple of \\( b - 1 \\) when \\( n \\) is expressed in base b. Hints: 15", "13.25 For all integer values of \\( n \\geq 2 \\), \\( k \\) will divide \\( n^3 - n \\). What is the greatest possible integer value of \\( k \\)? (Source: MATHCOUNTS) Hints: 98, 26", "13.27 Demonstrate that for natural numbers \\( m \\) and \\( n \\) that the modulo \\( 3^m \\) residue of \\( n \\) can be determined from its last \\( m \\) digits alone when expressed in base 12. Hints: 69", "13.28 Prove that a power of 2 cannot end in four equal digits. (Source: Mathematical Circles) Hints: 161" ], "candidateMarkers": [ { "id": "13.16", "label": null, "offset": 51 }, { "id": "13.17", "label": null, "offset": 191 }, { "id": "13.18", "label": null, "offset": 361 }, { "id": "13.19", "label": null, "offset": 430 }, { "id": "13.20", "label": null, "offset": 513 }, { "id": "13.21", "label": null, "offset": 933 }, { "id": "13.22", "label": null, "offset": 1061 }, { "id": "13.23", "label": null, "offset": 1194 }, { "id": "13.24", "label": null, "offset": 1362 }, { "id": "13.25", "label": null, "offset": 1547 }, { "id": "13.26", "label": null, "offset": 1722 }, { "id": "13.27", "label": null, "offset": 1821 }, { "id": "13.28", "label": null, "offset": 2020 }, { "id": "13.29", "label": null, "offset": 2126 }, { "id": "13.30", "label": null, "offset": 2292 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/page-0275.png", "sha256": "3f27697e2f77b68cb603affb37117bd6a73eff6e397981574773386ea362db19", "nextChapterHeadingPDF": 276 } }, { "source": "book-05", "pdfPage": 276, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a919dd84aa3fc149c7d1e50771303b8930ab014ba2699b8a41edfa0e2587b0e6", "textHash": "5ee1f2c640d114c829ddeb16cce3a581ae5f3c51f2d72127e03880c3db9a5647", "characters": 1225, "flags": [], "candidateHeadings": [ "14.1 Introduction", "Definition: A linear congruence equation is a congruence that involves a variable raised only to the first power." ], "candidateMarkers": [ { "id": "14.1", "label": null, "offset": 121 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0276.png", "sha256": "5d80ea24783fa3e58c57a64687cc1d961cab8278ff6b85820b9289bd8a9edefa", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 277, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9bc675790fe253af49e71ca66ee1da65433bbc1507e460fdc8b7a0edf7cffeec", "textHash": "d374c7366be49c08184aff2c0ea342f0476a00b15e1e2a8d7dd59da52f776201", "characters": 1962, "flags": [], "candidateHeadings": [ "14.2 Modular Inverses and Simple Linear Congruences" ], "candidateMarkers": [ { "id": "14.2", "label": null, "offset": 31 }, { "id": "14.1", "label": "Problem", "offset": 94 }, { "id": "14.2", "label": "Problem", "offset": 375 }, { "id": "14.3", "label": "Problem", "offset": 523 }, { "id": "14.4", "label": "Problem", "offset": 864 }, { "id": "14.5", "label": "Problem", "offset": 1241 }, { "id": "14.6", "label": "Problem", "offset": 1643 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0277.png", "sha256": "f58076a3eef3fdb7da246ef3749490d810ed83fea17f3847c3cbcd75db01afcf", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 278, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "cd3d3df4990eaf72f08ccbe99f1af2f4340e2e1a4101824bd1035c72dae7a388", "textHash": "fc7a624267baeb6375fe72544bcf23eb1479b5eba4ed7410006f48d05f9ffe2c", "characters": 2194, "flags": [], "candidateHeadings": [ "Concept: In much the same way that we solve ordinary algebraic linear equations, we solve simple linear congruences by manipulating both sides of the congruence until the variable is isolated." ], "candidateMarkers": [ { "id": "14.1", "label": "Problem", "offset": 53 }, { "id": "14.1", "label": "Solution for Problem", "offset": 324 }, { "id": "14.2", "label": "Problem", "offset": 1680 }, { "id": "14.2", "label": "Solution for Problem", "offset": 1744 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0278.png", "sha256": "ff85a20762832483b5ff4cb07f44157817acb9958cce749d5da5d6341a6ec9a1", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 279, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "205ec1cd26007cb2c27c6ebb9b56c0a0b7972921a6c79513ca2aa9121b845470", "textHash": "a8df836f94df611b4ed169d276793f83de093143c3947569f8108de4336f6e64", "characters": 1999, "flags": [], "candidateHeadings": [ "Definition: A modular inverse of an integer b (modulo m) is an integer \\(b^{-1}\\) such that" ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0279.png", "sha256": "f30700c657a1cd4a5f6e89cfda54e4c2377e245a2240b98acb117a5dfb6005c8", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 280, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8ebe2d2648c90525f45c99bf3dab3dd127a2d20b055a217e63fef24e5c9de88c", "textHash": "2d31334ba698cd8d57a1c8a2ec2c031dbe771ad9ce45e98a785738f86d967ccd", "characters": 2014, "flags": [], "candidateHeadings": [ "Important: If \\(gcd(b, m) > 1\\), then b does not have a modulo m inverse." ], "candidateMarkers": [ { "id": "14.3", "label": "Problem", "offset": 53 }, { "id": "14.3", "label": "Solution for Problem", "offset": 132 }, { "id": "14.4", "label": "Problem", "offset": 790 }, { "id": "14.4", "label": "Solution for Problem", "offset": 878 }, { "id": "14.5", "label": "Problem", "offset": 1773 }, { "id": "14.5", "label": "Solution for Problem", "offset": 1884 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0280.png", "sha256": "dec6234a2bc19fab637996f68a77c4282eafc501daab855873e9490b76ff28e7", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 281, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c676f9050fbe34a739a9de65c25fb94f0efc2db8f6ea07693e876735f75f7c7e", "textHash": "42e429b29550244225706050734f30a0e56638be5886dd15387ac83794ab8b35", "characters": 2789, "flags": [], "candidateHeadings": [ "Important: A modulo \\(m\\) residue \\(r\\) has a single inverse if and only if \\(gcd(m, r) = 1\\).", "Exercises" ], "candidateMarkers": [ { "id": "14.6", "label": "Problem", "offset": 1209 }, { "id": "14.6", "label": "Solution for Problem", "offset": 1475 }, { "id": "14.2.1", "label": null, "offset": 2375 }, { "id": "14.2.2", "label": null, "offset": 2583 }, { "id": "14.2.3", "label": null, "offset": 2652 }, { "id": "14.2.4", "label": null, "offset": 2700 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0281.png", "sha256": "54ae9a943abca0dae5b4e8b284234ef6df8eba1c3ec08a18b646f623237a2fec", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 282, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "43660e86b039b86c5f2e3d6752a8c308e236894623b90d735a7aa72a9a25bc06", "textHash": "14cf3e05b114e8dcde1a67642dc03696e71b90d3554dd2ac463e8a9385b7f678", "characters": 1603, "flags": [], "candidateHeadings": [ "14.3 Solving Linear Congruences" ], "candidateMarkers": [ { "id": "14.3", "label": null, "offset": 33 }, { "id": "14.7", "label": "Problem", "offset": 76 }, { "id": "14.8", "label": "Problem", "offset": 256 }, { "id": "14.9", "label": "Problem", "offset": 331 }, { "id": "14.10", "label": "Problem", "offset": 396 }, { "id": "14.11", "label": "Problem", "offset": 536 }, { "id": "14.12", "label": "Problem", "offset": 611 }, { "id": "14.13", "label": "Problem", "offset": 738 }, { "id": "14.7", "label": "Problem", "offset": 913 }, { "id": "14.7", "label": "Solution for Problem", "offset": 1093 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0282.png", "sha256": "2a3aaa89a8f91ccc49541641311c131febd2f8397a2f1d9c34eab58b0af8fdb0", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 283, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fd91ad5c5e5f4e1502c82e9fa0f7f4d39ba179bb15c9565dc7a3516cc0b7471a", "textHash": "d3cfc9442924af4c63ac53d729dc004da171ccac89ed3f4d6f78b9efd32d1393", "characters": 2098, "flags": [], "candidateHeadings": [ "Important: When the coefficient of a variable in a congruence is relatively prime with the modulus, we can get rid of the coefficient by multiplying both sides of the congruence by the inverse. This helps us isolate the variable." ], "candidateMarkers": [ { "id": "14.8", "label": "Problem", "offset": 767 }, { "id": "14.8", "label": "Solution for Problem", "offset": 840 }, { "id": "14.9", "label": "Problem", "offset": 1358 }, { "id": "14.9", "label": "Solution for Problem", "offset": 1421 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0283.png", "sha256": "2cc956c86c5ecc0894ae18ee2044d32d7765cd2f5fce1c6a802f4fc8251f3a63", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 284, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "7d1ccd41b76d8d0cfb9531ed04ba95ce9ea91b4cca61aaf7c2b3526017e810f3", "textHash": "f5174b2a4e29a0a90e9a60fe6000e6d2015b04d9fd43208c55c2cace4252397b", "characters": 2141, "flags": [], "candidateHeadings": [ "Important: A linear congruence \\(ax \\equiv b \\pmod{m}\\) has no solutions when \\(gcd(a, m) \\nmid b\\).", "Important: If \\(ak \\equiv b \\pmod{mk}\\) for integers \\(a\\) and \\(b\\), a positive integer \\(k\\), then" ], "candidateMarkers": [ { "id": "14.10", "label": "Problem", "offset": 900 }, { "id": "14.10", "label": "Solution for Problem", "offset": 1050 }, { "id": "14.11", "label": "Problem", "offset": 1549 }, { "id": "14.11", "label": "Solution for Problem", "offset": 1622 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0284.png", "sha256": "65558b42fcb264bad84ccc0e290eb85b9c643b2bf06158f55ac49cbb4181587e", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 285, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ed3a59785cead3968d6c696c7bcc8d0e575415fb9ffcc28eaa1d0189746400a2", "textHash": "d2a147d4bd2a41141853e93e2fcdb97ceefc84ba0ad6b3842d1f4ff327c93cad", "characters": 2514, "flags": [], "candidateHeadings": [ "Important: A linear congruence \\(ax \\equiv b \\pmod{m}\\) has solutions if and only if \\(gcd(a, m) \\mid b\\)." ], "candidateMarkers": [ { "id": "14.12", "label": "Problem", "offset": 245 }, { "id": "14.12", "label": "Solution for Problem", "offset": 368 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0285.png", "sha256": "97fea2e34a32e1ce01ddcf8da1c9cbb154e4035ccd2cdf93337e4f1cccf59e4b", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 286, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0284c55a60ff626dc5f0533efd198a07a5b4ae817960b2bbfd407740a64b500a", "textHash": "098a79b8f71ee22e8ce94a8431136631dbcf604aec93e152c44795dafeb38a74", "characters": 2368, "flags": [], "candidateHeadings": [ "Concept: Many remainder problems are just linear congruences." ], "candidateMarkers": [ { "id": "14.13", "label": "Problem", "offset": 474 }, { "id": "14.13", "label": "Solution for Problem", "offset": 569 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0286.png", "sha256": "3e3eebdc73c7f77dae9a59e8e52f92abf81444684f9026e7d964676f02a24bc3", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 287, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "df49fb6503b107fdbc655c8303678ae527b1c593bea336b8d33308b9fc070fa0", "textHash": "d3c8ebed16d35979570e4b8d1656a13517c5268f883b748cb04a258e33921c2e", "characters": 1488, "flags": [], "candidateHeadings": [ "Exercises", "14.4 Systems of Linear Congruences" ], "candidateMarkers": [ { "id": "14.3.1", "label": null, "offset": 656 }, { "id": "14.3.2", "label": null, "offset": 713 }, { "id": "14.3.3", "label": null, "offset": 774 }, { "id": "14.3.4", "label": null, "offset": 831 }, { "id": "14.4", "label": null, "offset": 893 }, { "id": "14.14", "label": "Problem", "offset": 1210 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0287.png", "sha256": "1b5c513caa5044f6d0d06912f235487cca2a65461d6704e55a68bf6aa737b6c9", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 288, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "25dda89915e64eb41f03d6ec5392879b2fd92323dd414639b3de57f9a73db113", "textHash": "6391d66cc70d815673c4e5d99d7df5ec554795839eef68194a204225ea72fbc1", "characters": 1062, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "14.15", "label": "Problem", "offset": 36 }, { "id": "14.16", "label": "Problem", "offset": 143 }, { "id": "14.17", "label": "Problem", "offset": 250 }, { "id": "14.18", "label": "Problem", "offset": 394 }, { "id": "14.19", "label": "Problem", "offset": 660 }, { "id": "14.14", "label": "Problem", "offset": 833 }, { "id": "14.14", "label": "Solution for Problem", "offset": 940 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0288.png", "sha256": "68716f89fc221c936182865b61ce2d22e939eae02d13c224d098207c826c97c3", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 289, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0edf25d2f2f3eae5b28d93f5285f4fa36e3c446c924e4e3f64656fda131ca830", "textHash": "844f80ab756d91a1147bbc442fe5b84ccf2a76b90b2b1b5a101a6d20912447b9", "characters": 1840, "flags": [], "candidateHeadings": [ "Concept: Looking at information in different mathematical forms gives us more ways to use it. When solving a system of linear congruences, we switch back and forth between using modular congruences and parametric equations." ], "candidateMarkers": [ { "id": "14.15", "label": "Problem", "offset": 1186 }, { "id": "14.15", "label": "Solution for Problem", "offset": 1298 }, { "id": "14.16", "label": "Problem", "offset": 1724 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0289.png", "sha256": "cccdca7188f7daa042b9d9af7b5c2b4a09053e7fe090d60ca1a377f2e7ee1ee3", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 290, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ac17b80e39056d37e0e0df4f1de72b82c82f4bd3988d375874886a893e79ac81", "textHash": "8863b284b1e350b272ff525d4af4335f5848c3731bf82189e654b4075dca5c44", "characters": 2076, "flags": [], "candidateHeadings": [ "Concept: We can think of our process of finding a single linear congruence that satisfies two linear congruences as simplifying a system of linear congruences. We solve systems of more than two linear congruences (of the same one variable) by continual simplification. We combine two congruences into one as many times as we need to either show that there are no solutions or to express the solutions as a single linear congruence." ], "candidateMarkers": [ { "id": "14.16", "label": "Solution for Problem", "offset": 36 }, { "id": "14.17", "label": "Problem", "offset": 838 }, { "id": "14.17", "label": "Solution for Problem", "offset": 950 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0290.png", "sha256": "53e92dd163f21be04d92d2e7dd22bf9114e2287f4aedd1fec90be8ce231ba45f", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 291, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "535386d2d5bb4f09729941b35f9547c88b227aea89eecb9ea7f4b31841fc7234", "textHash": "4f7a43b38a65fff1ba43ec90cca89c523b272c40cc5e26b9b184c6a056991628", "characters": 1651, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "14.18", "label": "Problem", "offset": 31 }, { "id": "14.18", "label": "Solution for Problem", "offset": 167 }, { "id": "14.19", "label": "Problem", "offset": 1123 }, { "id": "14.19", "label": "Solution for Problem", "offset": 1250 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0291.png", "sha256": "ec8322e1374358229d12fed596b74d944e828ef0bb2af26e4f021190b538a441", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 292, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c664197d54f8a4655398cbf8f8dd7d58dad24ea9ee5b4020c85819de4214f389", "textHash": "1397a1a42281083f2d4fec3bef265d6e16555a2cd08b343364ef21ee361a2a32", "characters": 1010, "flags": [], "candidateHeadings": [ "Exercises" ], "candidateMarkers": [ { "id": "14.4.1", "label": null, "offset": 496 }, { "id": "14.4.2", "label": null, "offset": 572 }, { "id": "14.4.3", "label": null, "offset": 650 }, { "id": "14.4.4", "label": null, "offset": 784 }, { "id": "14.4.5", "label": null, "offset": 914 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0292.png", "sha256": "d7e3308ab89d7ade36f5e32d9a1d0beb01854337f0b3d09f4ccc21f935145dac", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 293, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c8905486483632660a035ef0a74c9932f0c22d97d2675aa639738093de71b828", "textHash": "2205a0d4b3b5eb84bd1cef6fbb06154dd51cf69c799b3b95ef1ee19fbe9ac0dc", "characters": 1433, "flags": [], "candidateHeadings": [ "14.5 Summary", "Definitions:", "Important: A modulo m residue r has a single inverse if and only if \\(gcd(m, r) = 1\\). Otherwise, it has no inverse.", "Concept: In much the same way that we solve ordinary algebraic linear equations, we solve simple linear congruences by manipulating both sides of the congruence until the variable is isolated.", "Important: Switching between modular and parametric forms of information about integers helps us solve systems of linear congruences.", "Important: If \\(a \\equiv b \\pmod{m}\\) for integers a and b, a positive integer k, then", "Important: Let \\(a, b\\), and \\(c\\) be integers such as \\(a \\equiv bc \\pmod{m}\\). Then" ], "candidateMarkers": [ { "id": "14.5", "label": null, "offset": 31 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0293.png", "sha256": "3460ed4d2b82f04e99480547d844c8a1e6c580bffcd56c360750b7c941d02006", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 294, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "4e15c9ca248c521dfd658370109946b4555c75a10a99927bccb6ae5186d989b4", "textHash": "8923b5f34c4910a393a97b1b1a0c1ae067bcb6cc7887610b24da8e402b7da18a", "characters": 2440, "flags": [], "candidateHeadings": [ "Concept: Many remainder problems are just linear congruences.", "Concept: We can think of our process of finding a single linear congruence that satisfies two linear congruences as a simplification process. We solve systems of more than 2 linear congruences (of the same one variable) by continually simplifying. We combine two congruences into one as many times as we need to either show that there are no solutions or to express the solutions as a single linear congruence." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0294.png", "sha256": "cf4f558244c3907e05f13143fe2ed7d3a92b2fa5d39def10cdb5099c81bd8cac", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 295, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9f5da86e8adb77f5ede7378849650c893a847c617675216657a4f05b7e46fd1a", "textHash": "100b2cf9ddce9b56568a3fb60b83d039a97be6031ae20af75a4d63655be7be8b", "characters": 1078, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "14.20 Find the inverse of each of the following:", "14.21 Which modulo 20 residues have no inverse?", "14.22 Prove that in any modulus \\(m\\) that \\(m - 1\\) is its own inverse.", "14.23 Solve each of the following linear congruences.", "14.24 Solve the system of linear congruences:", "14.25 Solve the system of linear congruences:", "14.26 How many integers between 200 and 500 inclusive leave a remainder of 1 when divided by 7 and a remainder of 3 when divided by 4?", "14.27 Find the smallest positive integer that leaves a remainder of 5 when divided by 7, a remainder of 6 when divided by 11, and a remainder of 4 when divided by 13." ], "candidateMarkers": [ { "id": "14.20", "label": null, "offset": 48 }, { "id": "14.21", "label": null, "offset": 160 }, { "id": "14.22", "label": null, "offset": 209 }, { "id": "14.23", "label": null, "offset": 282 }, { "id": "14.24", "label": null, "offset": 437 }, { "id": "14.25", "label": null, "offset": 533 }, { "id": "14.26", "label": null, "offset": 649 }, { "id": "14.27", "label": null, "offset": 785 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0295.png", "sha256": "08dd7cb6b266e5ef22ec11e2483e90ea2b8e0a8c4305a56580e95478b542f932", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 296, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3425092699df0c8469c65a17870a75865045616ca4fe118fea89e895698c3f45", "textHash": "37aa504506c778f40c2151f2404e450d16cebdec307ab2bdaff2c7c0e0511a5a", "characters": 1639, "flags": [], "candidateHeadings": [ "CHALLENGE PROBLEMS", "Challenge Problems", "14.28 A natural number leaves a remainder of 7 when divided by 11 and a remainder of 10 when divided by 12. Find the remainder when the number is divided by 66.", "14.29 Let m be an integer greater than 101. Find the number of values of m such that 101 is its own inverse modulo m. Hints: 58", "14.30 Adam and Ben start their new jobs on the same day. Adam’s schedule is 3 workdays followed by 1 rest day. Ben’s schedule is 7 workdays followed by 3 rest days. On how many of their first 1000 days do both have rest days on the same day? (Source: AMC) Hints: 91" ], "candidateMarkers": [ { "id": "14.28", "label": null, "offset": 39 }, { "id": "14.29", "label": null, "offset": 201 }, { "id": "14.30", "label": null, "offset": 330 }, { "id": "14.31", "label": null, "offset": 597 }, { "id": "14.32", "label": null, "offset": 746 }, { "id": "14.33", "label": null, "offset": 955 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0296.png", "sha256": "544fea5b4114068f6d245780a688a866923f61ef0b9e914f17aeeb74079bbfcb", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 297, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8683ebf5b5e72ffc1374e6dea671ba28a41cabc708354910da6c20ad79fd6011", "textHash": "b47010c7d0076edcea8dd847fdc015c0b3e5e77d0195b9387df2f74cd808a647", "characters": 2152, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/page-0297.png", "sha256": "bcf10d21cb6fac7b412d6aa0dbbe8b0f7fad1675484bcdf97579f9491e1841a4", "nextChapterHeadingPDF": 298 } }, { "source": "book-05", "pdfPage": 298, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "94c105780c2d9f9535a4873d42248e73fa0ca9e760cf2652a9c85259ddae0e70", "textHash": "ab468c30ffd0b8fa159595d26fe19ad96947c5249160477504f28149d4c8353a", "characters": 1524, "flags": [], "candidateHeadings": [ "15.1 Introduction", "15.2 Familiar Factors and Divisibility" ], "candidateMarkers": [ { "id": "15.1", "label": null, "offset": 153 }, { "id": "15.2", "label": null, "offset": 1021 }, { "id": "15.1", "label": "Problem", "offset": 1426 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0298.png", "sha256": "f3dd35a02895cb16de734685b8b41cd3f3565d7b41b888080e16863f8fe58f71", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 299, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "13bd984f28c6f80a7f1f744ee023c9abc383881e7be7cba96fca39cb7fb22a37", "textHash": "e7307ee2e238f28189420eb49439744d7cf84c1003a7dd88be06ce64c65ebbfa", "characters": 1718, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "15.2", "label": "Problem", "offset": 25 }, { "id": "15.3", "label": "Problem", "offset": 110 }, { "id": "15.4", "label": "Problem", "offset": 224 }, { "id": "15.5", "label": "Problem", "offset": 357 }, { "id": "15.6", "label": "Problem", "offset": 505 }, { "id": "15.7", "label": "Problem", "offset": 577 }, { "id": "15.1", "label": "Problem", "offset": 747 }, { "id": "15.1", "label": "Solution for Problem", "offset": 843 }, { "id": "15.2", "label": "Problem", "offset": 1258 }, { "id": "15.2", "label": "Solution for Problem", "offset": 1343 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0299.png", "sha256": "f8522d0f9f14bd278434de110cda5db61c12ce54c6ec95705f81efaa1b0e45a7", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 300, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "6c858a7e9719ffb740edc58b9a2a5c2cf22f3e7399dd48a0ff24caf300d3a4e1", "textHash": "f398c7957ad8d791f6d561d0483caedf294b156eba5313031ae8fd9a0212adb0", "characters": 2739, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "15.3", "label": "Problem", "offset": 775 }, { "id": "15.3", "label": "Solution for Problem", "offset": 889 }, { "id": "15.4", "label": "Problem", "offset": 2601 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0300.png", "sha256": "06d07f86243ee5d18b55706bf2a59c9eab73498e86dc4d69ba83bd4bacd34e1a", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 301, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "fdd5cf2210f0daa4bdb77b40624c6c2b3061f18b403189ce09548bfd90f7d247", "textHash": "8305068e569a0e28e3c4044448a5c5b21e005a464ab31f1fc7f02f7120e4243b", "characters": 2224, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "15.4", "label": "Solution for Problem", "offset": 25 }, { "id": "15.5", "label": "Problem", "offset": 1250 }, { "id": "15.5", "label": "Solution for Problem", "offset": 1399 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0301.png", "sha256": "620fc68d1d42f01bf189417c195b1cd7fa3765cdb7b405405cde79acf0dd3e9a", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 302, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d44b881f8f51f4ebdacd1093e1c2e5066169ddcbdc36de0004ae3105c03751b1", "textHash": "15223f9ffe01bc57636fbeee9ecf157291f5e9054425a1e87e3492fb1594f843", "characters": 2110, "flags": [], "candidateHeadings": [ "Concept: Number sense is not just about solving problems. Number sense helps us connect mathematics in new and different ways. Sometimes this means finding quick ways to check our work. Sometimes it means noticing entirely new mathematical relationships.", "15.3 Algebraic Methods of Arithmetic" ], "candidateMarkers": [ { "id": "15.6", "label": "Problem", "offset": 270 }, { "id": "15.6", "label": "Solution for Problem", "offset": 343 }, { "id": "15.7", "label": "Problem", "offset": 932 }, { "id": "15.7", "label": "Solution for Problem", "offset": 1102 }, { "id": "15.3", "label": null, "offset": 1776 }, { "id": "15.8", "label": "Problem", "offset": 2055 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0302.png", "sha256": "bc9963f817b17d12a2239949e946f2f57360534466f88d788d72139a034fe6cf", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 303, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d8089f057d63ceeea5d5030341077307a9f8d844474a253c8ad6757db545985d", "textHash": "0a6b3d7fedef9fe8bbc9a4eff0862b836673a1fb41e8a50ebf49464a588970c6", "characters": 2086, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "15.9", "label": "Problem", "offset": 25 }, { "id": "15.10", "label": "Problem", "offset": 85 }, { "id": "15.11", "label": "Problem", "offset": 161 }, { "id": "15.12", "label": "Problem", "offset": 220 }, { "id": "15.8", "label": "Problem", "offset": 347 }, { "id": "15.8", "label": "Solution for Problem", "offset": 398 }, { "id": "15.9", "label": "Problem", "offset": 1327 }, { "id": "15.9", "label": "Solution for Problem", "offset": 1387 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0303.png", "sha256": "3fd1696a0ad0cef8aeed5181ba5512dbf21e51a2ce030333fc9d9da20c24abfd", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 304, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0fc25cc5ab8bdb02d0d402be66be9aff1a6ab0bdf10a54a39944d9320957e84f", "textHash": "99c5bfed5fc474095a27dbb87cf5589c48a70b48829e47f89eedf04f1e9b0b43", "characters": 2048, "flags": [], "candidateHeadings": [ "Important: It helps to know your primes up to 100 because they are common in practical number theory problems.", "Concept: The more time you spend thinking about and solving number theory problems, the more quickly you can see useful relationships between integers. Students develop number sense more from practice than natural ability.", "Concept: Awareness of the ways in which algebra works together with number theory (or other subjects) adds significantly to our ability to use the tools of both algebra and number theory to solve problems." ], "candidateMarkers": [ { "id": "15.10", "label": "Problem", "offset": 701 }, { "id": "15.10", "label": "Solution for Problem", "offset": 760 }, { "id": "15.11", "label": "Problem", "offset": 1404 }, { "id": "15.11", "label": "Solution for Problem", "offset": 1463 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0304.png", "sha256": "6016d2890750589ac97c8af04b4afc102646b62c49cdb7d545129ca4ffd9a127", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 305, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "23beb08cb887c49e44256250c487655d70053fdf8691a8d472259e397ccaed77", "textHash": "46ba09c5f4fd346cde769729212e0c77e2b5b7b016e7e69befe2eae4b0f3557e", "characters": 1807, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "15.12", "label": "Problem", "offset": 560 }, { "id": "15.12", "label": "Solution for Problem", "offset": 687 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0305.png", "sha256": "91934f661b84fc112c6daca411fa016135a3699d1bf7b13de7c13ea3d92f56af", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 306, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "8cc8ded186a8ce42f3144637263d3f83dd815dca6a4f62a987e3e2b0f0e7b1b1", "textHash": "746b4e16fbcb0b3a084086332937b81573acad3ec87ec2973d89fed09ef61f52", "characters": 2198, "flags": [], "candidateHeadings": [ "Concept: Working through the long solution to a problem helps us better understand why the quick solution works and increases the range of our problem solving abilities." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0306.png", "sha256": "d5871b2f63fbee641d8f1752a7d953ce8fa4cc1cc78826e93a6d0176d55e7777", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 307, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "87e0e376aed58fad3dd26857f8d2d4c6ec3901549da8638a217f95a5b4cc7b0f", "textHash": "1a1ec2695676dd412cb9137b55acf738285dcc05e269eaa9e1d21e59408fed93", "characters": 1917, "flags": [], "candidateHeadings": [ "15.4 Useful Forms of Numbers" ], "candidateMarkers": [ { "id": "15.4", "label": null, "offset": 25 }, { "id": "15.13", "label": "Problem", "offset": 560 }, { "id": "15.14", "label": "Problem", "offset": 734 }, { "id": "15.13", "label": "Problem", "offset": 823 }, { "id": "15.13", "label": "Solution for Problem", "offset": 997 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0307.png", "sha256": "e26af29dcd6ab336b3336da06aa7db2fbb1bcc4dd6b2544635aff79cf155a02f", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 308, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "a7c2c90c887edc4e9966d69a3fd8b718c5166989ab31d4d813c558b4edb42859", "textHash": "7a83b063aed5973433f0f67ae51ddf84bbb3e120402e3efc101261ed5f97035f", "characters": 2197, "flags": [], "candidateHeadings": [ "Concept: Change numbers to forms that are most useful in a given problem.", "Concept: While we may be most comfortable with a particular method of problem solving, awareness of other methods often helps us solve more difficult problems with relative ease." ], "candidateMarkers": [ { "id": "15.14", "label": "Problem", "offset": 884 }, { "id": "15.14", "label": "Solution for Problem", "offset": 973 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0308.png", "sha256": "e87ef35316c2121fbd96c26981f24030fba4ef65f35651abfa001dd92b2df2a2", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 309, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "1848f7d17242c3ef33a3bd1da5288ffea8d50326e0c486ab56f615bcb90cb9df", "textHash": "22ee550bf8a780e99e9d66c32677cc05d1704290df11f24a58ffc3b598dbd9db", "characters": 2235, "flags": [], "candidateHeadings": [ "15.5 Simplicity" ], "candidateMarkers": [ { "id": "15.5", "label": null, "offset": 25 }, { "id": "15.15", "label": "Problem", "offset": 407 }, { "id": "15.16", "label": "Problem", "offset": 533 }, { "id": "15.17", "label": "Problem", "offset": 631 }, { "id": "15.18", "label": "Problem", "offset": 776 }, { "id": "15.19", "label": "Problem", "offset": 882 }, { "id": "15.15", "label": "Problem", "offset": 963 }, { "id": "15.15", "label": "Solution for Problem", "offset": 1089 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0309.png", "sha256": "1214178406bfdc016c7b1a689a72e746e72445871602f2306bb670da0e92cdb2", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 310, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "64edbbb390d435c2829a3fe2422995c2bfce81385815478d7278ad4cfe5ed3bd", "textHash": "cf462c8f385dcf2b5ba878bbd08df7fcffb346b19da313f6f3c43ee6c961c7e7", "characters": 1969, "flags": [], "candidateHeadings": [ "Definition: The property of an integer being even or odd is known as parity.", "Concept: Parity considerations are really just a matter of working in modulo 2." ], "candidateMarkers": [ { "id": "15.16", "label": "Problem", "offset": 721 }, { "id": "15.16", "label": "Solution for Problem", "offset": 819 }, { "id": "15.17", "label": "Problem", "offset": 1298 }, { "id": "15.17", "label": "Solution for Problem", "offset": 1443 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0310.png", "sha256": "48d7c6587e1f9b01125789a0177d3835bd6608c00b6462ce873782c3daf80dd7", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 311, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "304e00a2173772d900d8194432bc0acecac6c064efeacecc7c37bea0dc5069db", "textHash": "374be07546bf3ae09680004c1c1e9510562715ad5cd4c8c0bf1a60f36d81063b", "characters": 2112, "flags": [], "candidateHeadings": [ "Concept: Don't make a problem harder than it needs to be. While experienced problem solvers have learned that hunting through lists is often an extremely slow way to solve a problem, there are exceptions. If you can narrow your search to a small list, hunting through that list might be easiest." ], "candidateMarkers": [ { "id": "15.18", "label": "Problem", "offset": 662 }, { "id": "15.18", "label": "Solution for Problem", "offset": 768 }, { "id": "15.19", "label": "Problem", "offset": 1718 }, { "id": "15.19", "label": "Solution for Problem", "offset": 1799 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0311.png", "sha256": "be467f3d5694fb620e67a75a35f53dad715480fb27e0e8445e982690dc938faf", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 312, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2964e7b80e0874d2b15e25754900f34575799859fd6fe96eaa769a2526521eaa", "textHash": "708a42ea252984eae541092e249e6ff3ddeb7985572ba957b8da67e2deebf2d0", "characters": 2143, "flags": [], "candidateHeadings": [ "Concept: We can often use clues to gather enough information to solve a problem that has no obvious or direct solution.", "15.6 Summary", "Concept: Number sense is not just about solving problems. Number sense helps us connect mathematics in new and different ways. Sometimes this means finding quick ways to check our work. Sometimes it means noticing entirely new mathematical relationships.", "Important: It helps to know your primes up to 100 because they are common in practical number theory problems.", "Concept: The more time you spend thinking about and solving number theory problems, the more quickly you can see useful relationships between integers. Students develop number sense more from practice than natural ability." ], "candidateMarkers": [ { "id": "15.6", "label": null, "offset": 671 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0312.png", "sha256": "7aa791de26f13f2ae8b862f09cb65bb4b092ac7c1ea3dd8df4fac6f60f03de6e", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 313, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9a15556ee5fbc4329227530f3547c50c5dcdedd61c2c2d751e3dc8fd623ce2ed", "textHash": "c880da758cea0e2a5a6eb56c315fde7bab0103602a7673bf16658ee50ab5713d", "characters": 2851, "flags": [], "candidateHeadings": [ "Concept: Awareness of the ways in which algebra works together with number theory (or other subjects) adds significantly to our ability to use the tools of both algebra and number theory to solve problems.", "Concept: Working through the long solution to a problem helps us better understand why a quick solution works and increases the range of our problem solving abilities.", "Concept: When the forms of numbers we are working with do not seem to give way to a practical solution, changing the form in which we express those numbers to match the problem often simplifies the problem greatly.", "Concept: While we may be most comfortable with a particular method of problem solving, awareness of other methods often helps us solve more difficult problems with relative ease.", "Concept: Parity considerations are really just a matter of working in modulo 2. Parity arguments are common enough in problem solving to give their use special attention (in addition to their own name).", "Concept: Don't make a problem harder than it needs to be. While experienced problem solvers have learned that hunting through lists is often an extremely slow way to solve a problem, there are exceptions. If you can narrow your search to a small list, hunting through that list might be easiest." ], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0313.png", "sha256": "70b984bf2c5176e9faa2dda628fad4ee1f409d1aeacddd4a5e5bb0531c7b062e", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 314, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "16098d8d607cda50c26b4574b9b49a90af594888fb90c9194adb1e3e3fe29f6f", "textHash": "9e37a8c69a273ad9d6a24ef50fda0df7a078fd4644361594d1c57bead620b981", "characters": 2131, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "Concept: While there may not be a direct way to compute the answers to some problems, we can often use clues to gather enough information to solve the problem.", "REVIEW PROBLEMS", "15.20 What number is one-half of one-tenth of one-fifth of one-half of one million?", "15.21 Find the smaller of two prime numbers whose sum is also prime.", "15.22 There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5, and 7 exactly once. Only one of the these four-digit numbers is a multiple of another one. Which one is it? (Source: AMC)", "15.23 Compute 55 · 1212 - 15 · 1212.", "15.24 If \\(2^5 \\cdot 8^3 \\cdot 16^2 = 4^m\\), what is the value of \\(m\\)?", "15.25 What is the least natural number, greater than 1, that is a factor of 11000 + 1100 + 11?", "15.26 How many digits are in the product \\(8^{12} \\cdot 25^{8}\\)?", "15.27 Find the smallest positive multiple of 1999 that ends in 2006 (last four digits).", "15.28 How many ordered triples of three prime numbers exist for which the sum of the members of the triple is 24? (Source: MATHCOUNTS)", "15.29 What is the largest 4-digit number that is equal to the cube of the sum of its digits?", "15.30 Find the value of \\(61^3 - 3 \\cdot 61^2 + 3 \\cdot 61 - 1\\).", "15.31 What is the smallest whole number such that, when divided by each of 10, 9, 8, 7, ..., 2, gives a remainder of 9, 8, 7, 6, ..., 1, respectively? (Source: MATHCOUNTS)", "15.32 Find the greatest common divisor of 792 and 39654.", "15.33 The positive integers \\(A\\), \\(B\\), \\(A - B\\), and \\(A + B\\) are all prime numbers. Find the sum of these four primes. (Source: AMC)" ], "candidateMarkers": [ { "id": "15.20", "label": null, "offset": 689 }, { "id": "15.21", "label": null, "offset": 795 }, { "id": "15.22", "label": null, "offset": 865 }, { "id": "15.23", "label": null, "offset": 1074 }, { "id": "15.24", "label": null, "offset": 1112 }, { "id": "15.25", "label": null, "offset": 1186 }, { "id": "15.26", "label": null, "offset": 1303 }, { "id": "15.27", "label": null, "offset": 1370 }, { "id": "15.28", "label": null, "offset": 1459 }, { "id": "15.29", "label": null, "offset": 1595 }, { "id": "15.30", "label": null, "offset": 1689 }, { "id": "15.31", "label": null, "offset": 1756 }, { "id": "15.32", "label": null, "offset": 1929 }, { "id": "15.33", "label": null, "offset": 1987 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0314.png", "sha256": "16e119c4cdcbb76c62257c0a114f96774479f6f7cc3eb704b6828dae7105d65c", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 315, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "dba58f6c584dc27014fed71ff9dac8d4b9e3267ce78a1dcfe2ad8ab68a4c7704", "textHash": "2b8baf983c1f43fe332f0b226a3d359af4f23739c153bb13dac8d5c88f900355", "characters": 2312, "flags": [], "candidateHeadings": [ "15.34 In how many different ways can 125 be written as the sum of 3 positive perfect squares if order does not matter and the squares do not need to be distinct?", "15.35 What is the value of \\(\\sqrt{(27)(243)}\\)?", "15.36 Find the sum of the total number of digits in each of \\(2^{2006}\\) and \\(5^{2006}\\).", "15.37 Find the smallest positive integer that is one less than a multiple of each of 3, 5, 7, 9, and 11.", "15.38 Use the digits 2, 4, 5 and 7 each once to create two prime numbers. What is the smallest possible product of two prime numbers created from these digits? (Source: MATHCOUNTS)", "15.39 Evaluate", "15.40 What is the sum of all positive two-digit integers for which the sum of the digits is a multiple of 5 and the remainder is 5 when divided by 7? (Source: MATHCOUNTS)", "15.41 If the integer 152AB1 is a perfect square, what is the sum of the digits of its square root?", "15.42 Evaluate", "15.43 How many positive integers less than 100 leave a remainder of 1 when divided by 2 and also when divided by 3? (Source: Mandelbrot)", "15.44 The sum of two prime numbers is 20. What is the greatest possible product of these two primes? (Source: MATHCOUNTS)", "15.45 What is the value of \\(5^2 - (10)(6) + 6^2\\)?", "15.46 Find the smallest four-digit multiple of 5 that has 2, 4, and 8 among its digits.", "15.47 Compute A divided by B where \\(A = 66666^4\\) and \\(B = 22222^4\\).", "15.48 Find the prime factorization of \\(5^8 - 16\\).", "15.49 What is the remainder when the product of the first 25 prime numbers is divided by 4? (Source: MATHCOUNTS)", "15.50 If \\(2^{20} - 2^{19} = 2^x\\), what is the value of \\(x\\)?", "15.51 There is more than one integer greater than 1 which, when divided by any integer k such that \\(2 \\leq k \\leq 11\\), has a remainder of 1. What is the difference between the two smallest such integers? (Source: AHSME)", "15.52 Find the greatest common divisor of 8008 and 14014.", "15.53 Let \\(D(a, b, c)\\) denote the number of multiples of \\(a\\) that are less than \\(c\\) and greater than \\(b\\). For example, \\(D(2, 3, 8) = 2\\) because there are two multiples of 2 between 3 and 8. What is \\(D(9^3, 9^4, 9^6)\\)? (Source: UNCC)", "15.54 Evaluate" ], "candidateMarkers": [ { "id": "15.34", "label": null, "offset": 25 }, { "id": "15.35", "label": null, "offset": 188 }, { "id": "15.36", "label": null, "offset": 238 }, { "id": "15.37", "label": null, "offset": 330 }, { "id": "15.38", "label": null, "offset": 436 }, { "id": "15.39", "label": null, "offset": 618 }, { "id": "15.40", "label": null, "offset": 650 }, { "id": "15.41", "label": null, "offset": 822 }, { "id": "15.42", "label": null, "offset": 922 }, { "id": "15.43", "label": null, "offset": 981 }, { "id": "15.44", "label": null, "offset": 1119 }, { "id": "15.45", "label": null, "offset": 1242 }, { "id": "15.46", "label": null, "offset": 1295 }, { "id": "15.47", "label": null, "offset": 1384 }, { "id": "15.48", "label": null, "offset": 1457 }, { "id": "15.49", "label": null, "offset": 1510 }, { "id": "15.50", "label": null, "offset": 1624 }, { "id": "15.51", "label": null, "offset": 1689 }, { "id": "15.52", "label": null, "offset": 1912 }, { "id": "15.53", "label": null, "offset": 1971 }, { "id": "15.54", "label": null, "offset": 2217 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0315.png", "sha256": "16aeebe884812fddc8467cd082365c9bf1ec33ac8afc55b20042c5f8d346d41d", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 316, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "c42ca667911032ed6c6d8661595be78c189e3f9cc69c8cc2be7590b71378ff3b", "textHash": "0d1f18ce7c051c23042b58692b1c65bb994839ffc519bd00a94e30c98c7e51d8", "characters": 1240, "flags": [], "candidateHeadings": [ "REVIEW PROBLEMS", "15.55 Perform the following calculation entirely in your head:", "15.56 Find positive integers \\((a, b, c, d)\\) such that", "15.57 Calculate", "15.58 Find the value of \\(\\frac{22004 + 22001}{22003 - 22000}\\).", "15.59 Perform the following calculation entirely in your head:", "15.60 Given that \\(a\\) and \\(b\\) are natural numbers such that", "15.61 Find the least positive integer \\(n\\) for which \\(\\frac{n - 13}{5n + 6}\\) is a non-zero reducible fraction.", "15.62 The numbers from 1 to 2002 are listed in the following order: First all numbers that are not divisible by 3 are listed in (increasing) order. Then all numbers that are divisible by 3 but not by \\(3^{2}\\) are listed in order. Then all numbers that are divisible by \\(3^{2}\\) but not by \\(3^{3}\\) are listed in order, etc. What is the last number in the list? (Give the entire number, not just its last digit.)" ], "candidateMarkers": [ { "id": "15.55", "label": null, "offset": 16 }, { "id": "15.56", "label": null, "offset": 102 }, { "id": "15.57", "label": null, "offset": 227 }, { "id": "15.58", "label": null, "offset": 269 }, { "id": "15.59", "label": null, "offset": 335 }, { "id": "15.60", "label": null, "offset": 447 }, { "id": "15.61", "label": null, "offset": 602 }, { "id": "15.62", "label": null, "offset": 734 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0316.png", "sha256": "c6c31bd002cb2bd7e42734563609caecb2f171ce8536d40d3d9b05899876b70f", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 317, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ddc3e89da3c93c7faad5b6f4d1f33a87b8a3f50ce3a44a131c9e5beab44c0655", "textHash": "ef220b54193ef854613bcca20c76c43c8808c997509db109301503cb4e2266c3", "characters": 1751, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/page-0317.png", "sha256": "95896df607cef66f6a1bd6301d2c2e6b7153db2715b8e6c79c916c64bcced47a", "nextChapterHeadingPDF": 318 } }, { "source": "book-05", "pdfPage": 318, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2fc6bd0632ed41c2edbd169748856fa6fcfcbd461b0abb748c106f21e98d3b1e", "textHash": "ed191bb3345a0dcb8bef058c1bc6cc03a4911a4bd4e2de01966de8f4475165da", "characters": 2097, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0318.png", "sha256": "f4998bf78b87119cff4662196b4a52dbf75f007fed13de85e1e314e637459c8d" } }, { "source": "book-05", "pdfPage": 319, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "9f500d49c8df65662d8cdd4cc478fed22d41519e84d1072be555e1f5fd43b6fc", "textHash": "9609fa02cf6c3ad5f58d1af6b04011a50411f87949df58e18a6571e90550fc4a", "characters": 2229, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0319.png", "sha256": "6153dfdda5cb34f6974677acd202600127c08eeff51422a327b04c1b39b6ad90" } }, { "source": "book-05", "pdfPage": 320, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "2d804379a67244dd2f08504a2579a1165b5a1c89248eba7774932c4c3f296d5a", "textHash": "b34bc48f8e38e5ad660a5cb74b9b197cdff64d68f0160d6d48a9541c9f77f0c7", "characters": 1892, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0320.png", "sha256": "e9b146c147f6c693626c7d4453a1212f614407fe4159e00d6d6f974a48832bfd" } }, { "source": "book-05", "pdfPage": 321, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "3faa00f5d1d76f464193e53e17b5b13fa47cd674e6d696c9c4c3b11e994c271f", "textHash": "46dcb5ba6bce7bcbe347fbfc12b3338fa8cbd181c7ee26cea065b13bd4d8eeda", "characters": 2390, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0321.png", "sha256": "5edae870e3a5129600eaf4e62ae60c5abbb4d2969a5a1901b37966abaa333eb3" } }, { "source": "book-05", "pdfPage": 322, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "d7355243e3218fdf347a2b5f1b6b1ea49bb60f85524fd0e315cecf3b147f76ec", "textHash": "d2660a5232888190f0169c71e4f8c7ea09215a7e4c5b5132cf88b37c9d1e4ad4", "characters": 2103, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0322.png", "sha256": "a8bc8a02928d743dcb478ee4962dc34a96ba747443bf90e0a2e1887afbce0513" } }, { "source": "book-05", "pdfPage": 323, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "0f0e763c6f3af12add37753505d8adbede6308557fb567e71a70ceb86a631128", "textHash": "02ec5bfb92adabf07b5290fefa020e185ecfd7ad57a1917872c3e5abb22e2a2b", "characters": 2111, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0323.png", "sha256": "b825ffd11904fa71d073296ca6819a770757a37c101fbc165c13b4a28ce6f822" } }, { "source": "book-05", "pdfPage": 324, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "239b7e16e0607217a8330d0e79e57550ec7335b3b3a212e763ccc411e8cee38a", "textHash": "06b5c6f77d1ee0843ad1d5e66205fe25257daa11692a68ee6b5bef6b1efbb1cc", "characters": 1957, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0324.png", "sha256": "d8b6e42d71f47c8cb91dc67dfc27ba8dcf365a3520e72a092ae379977f302682" } }, { "source": "book-05", "pdfPage": 325, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "cf498b710c65f70cbe7d72bb5da3b0bfb9836bd3a4a371fc9583b9cc6a97be04", "textHash": "e3e34d0270d88e1ab8c3a87a039a0b31f953766e3fb98e28afd1e73e35d751a9", "characters": 1149, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "selected-hints", "evidence": "book05-source/page-0325.png", "sha256": "55f9ec76854013a9c8ecc39af6e8b86dcab1fcb97d9ced29c2d34c9547655fb2" } }, { "source": "book-05", "pdfPage": 326, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "ac5605a576b3fe97bd913163bd71f5b8fc5493f54716d77633394d744880c8cc", "textHash": "4f0b1c22e8492c01408c113cf222f7ef1b33f694b7c594165b6c3fcc9bd7b0aa", "characters": 1609, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "index-navigation", "evidence": "book05-source/page-0326.png", "sha256": "9f7452d560eb3a5dcca51c5c222c8d0cd341d7bcad827a8dbcac02f927c1692b" } }, { "source": "book-05", "pdfPage": 327, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "acde752a8df0d7d6b663b12b810766a9fb257b7534a02d2b48665faf2d37ecd2", "textHash": "8e993a91eb0eaf8e12c85abeaa0e6c0081563a54824e2c2a802ccbca8cddfe64", "characters": 2092, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "index-navigation", "evidence": "book05-source/page-0327.png", "sha256": "71ab3ecd9ee14a30752a12a5f35b6329527614381c8ecc8f966f21c8c5bca4a5" } }, { "source": "book-05", "pdfPage": 328, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "60291025877d2685dd54c024d32687f711b3f99d3dd0e578776107d7b773d6ca", "textHash": "b10451f55d188ade190b45d3106c2626c8d06163cc3e835f0eae36da5deb811d", "characters": 2005, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "index-navigation", "evidence": "book05-source/page-0328.png", "sha256": "fd624b33723fe815ee9cc50b0520a9bd470adf947185385e1e0d0f6e1d5f7020" } }, { "source": "book-05", "pdfPage": 329, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "54c960016e69779ad0bf8b0c483fa627038955469837b3b90637b86a43ecbd7a", "textHash": "968c2b7276968289a5bc821387f9d7fadd4fe84a947d0fa42f78575b99af756e", "characters": 720, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "index-navigation", "evidence": "book05-source/page-0329.png", "sha256": "295a66988a82649a90a6ed7373a9f659e95509708f54f03a03c924b7c4d1b80b" } }, { "source": "book-05", "pdfPage": 330, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory.pdf", "ocrHash": "5ebe9befacf71a3b4e94b6511d51cc67af1fc14cfecab1fbd85fbace40ec55da", "textHash": "392bf28f906cb863f051f3f7fd497e7c2911ace96d1f0d1c0ffc50beb57099ff", "characters": 1622, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/page-0330.png", "sha256": "cd1500df1b4fba176daa835a3e12aefff574fbbcf0bf2e5bc4937fcd4a0679bb" } }, { "source": "answers-05", "pdfPage": 1, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a6e55d7248bc5e253ae89acd97dd428ae4daacb5d8374a85ba9c8ad6b22619b0", "textHash": "70f6693c129e2735dca824739322fbf6b2bda282416047cfab5a0bd99359fcf2", "characters": 89, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0001.png", "sha256": "260b361b22bea24e29f31d7f0145997622811c28acc455d31b7e89909ec7fd52" } }, { "source": "answers-05", "pdfPage": 2, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "53571e3410e5f3bc7b218f7aee10f0a8f05f17741c66030708ebe8fc182c916d", "textHash": "fe3afb6cc46db1fcaba79319f8abf94bcbbc3484da088a8b5874877f16be4ec1", "characters": 99, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0002.png", "sha256": "880ca7edef181db2ba40f8a15dbc2da5f6907bbd7c7a61e2f39729cdbd1d44c1" } }, { "source": "answers-05", "pdfPage": 3, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2e1f1298cf74f097fe5b419f70cabffdf3fe99fbe07a6c147d577700108aebbe", "textHash": "ff087ed755f263a4f84f08f5e2863f0a4094d431497927784084d57a4003b567", "characters": 1078, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0003.png", "sha256": "b249d27cdd4a3820131558ed9b23d8d55e7bb1235a9be342aa444fe7ae35ec43" } }, { "source": "answers-05", "pdfPage": 4, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "952ea0aa71b2fbe8331c936fbfd87b0a99dc0cfd1508db2827da296d588a4dae", "textHash": "45e7726f2757ff1fc6b26cb7eba75a0573eda469f14db9dbd961ff9fb757e421", "characters": 230, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0004.png", "sha256": "884862dc2a51034aed423966555c547538fb541bfd09cc477a34b56f416597ee" } }, { "source": "answers-05", "pdfPage": 5, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "632eb64af8062868085fb11bf28748556deccf6c69b12fc05b14084992be5894", "textHash": "a23361eedaab2c999d0500ec0ad442ef02b0f1543bac5fd5a221a2f39efb1bde", "characters": 209, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0005.png", "sha256": "888a75a219705f9c2e116bd3fd1231f0dddc32b7a223ecbfa81269bc42ce53ce" } }, { "source": "answers-05", "pdfPage": 6, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "e438d3057e61d3393a10e40b5157ce5b486bcc7c2e644db303e191319d5ddb7a", "textHash": "94c7139275d30363040adff5ea645335e9b10c2b3912cdddc1c0063b1cc3933a", "characters": 1525, "flags": [], "candidateHeadings": [ "Exercises for Section 1.2", "Exercises for Section 1.3" ], "candidateMarkers": [ { "id": "1.2.1", "label": null, "offset": 59 }, { "id": "1.2.2", "label": null, "offset": 176 }, { "id": "1.2.3", "label": null, "offset": 492 }, { "id": "1.2.4", "label": null, "offset": 563 }, { "id": "1.3.1", "label": null, "offset": 794 }, { "id": "1.3.2", "label": null, "offset": 865 }, { "id": "1.3.3", "label": null, "offset": 963 }, { "id": "1.3.4", "label": null, "offset": 1065 }, { "id": "1.3.5", "label": null, "offset": 1151 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0006.png", "sha256": "47645f032ef4f319d05617a9f3e2be457e16ed35214790cf0d2b70c8d3f48c68", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 7, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "1f1fe80c1117a69ad05f0e62f96f272f61601cf2e21f67faa961057dc11218d8", "textHash": "a62b54d8e38688060da285a16a3ab15d140a95bc136770499a9ea8152bb7b025", "characters": 2487, "flags": [], "candidateHeadings": [ "Exercises for Section 1.4" ], "candidateMarkers": [ { "id": "1.3.6", "label": null, "offset": 32 }, { "id": "1.3.7", "label": null, "offset": 317 }, { "id": "1.3.8", "label": null, "offset": 421 }, { "id": "1.3.9", "label": null, "offset": 645 }, { "id": "1.3.10", "label": null, "offset": 859 }, { "id": "1.4.1", "label": null, "offset": 1109 }, { "id": "1.4.2", "label": null, "offset": 1788 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0007.png", "sha256": "f801d61426a8e518e3c709472131fbdf0045dc6d92b44e0337da7770d673393d", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 8, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a08e1d4ead0dd075d4743aa9b329a695e0b6037da559358a22c0f37251d1e8cf", "textHash": "bb861325278aeb6a56fd12332b3292b3ae857d274a90c6c733e23a8b6a0be279", "characters": 1329, "flags": [], "candidateHeadings": [ "Exercises for Section 1.5", "Exercises for Section 1.6" ], "candidateMarkers": [ { "id": "1.4.3", "label": null, "offset": 12 }, { "id": "1.4.4", "label": null, "offset": 101 }, { "id": "1.4.5", "label": null, "offset": 161 }, { "id": "1.4.6", "label": null, "offset": 223 }, { "id": "1.4.7", "label": null, "offset": 313 }, { "id": "1.4.8", "label": null, "offset": 377 }, { "id": "1.4.9", "label": null, "offset": 471 }, { "id": "1.5.1", "label": null, "offset": 565 }, { "id": "1.6.1", "label": null, "offset": 790 }, { "id": "1.6.2", "label": null, "offset": 1007 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0008.png", "sha256": "e90086565de238c830e74098288bb531f07853b8a15e2311f407f8d3f13f2399", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 9, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "17e9e2020f1b0400e542d82d0d4acd148defb1763ce5f207fd7f198ddec54af3", "textHash": "4907238ae874eb9d46ed96822b634caba38e89d1a84d7a6bfb3a6f9842091c09", "characters": 1897, "flags": [], "candidateHeadings": [ "Exercises for Section 1.7", "Review Problems" ], "candidateMarkers": [ { "id": "1.7.1", "label": null, "offset": 58 }, { "id": "1.11", "label": null, "offset": 978 }, { "id": "1.12", "label": null, "offset": 1068 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0009.png", "sha256": "34130ae77e96622646e06e22c17665f2014022a1cc0ce2860289b3db0476e902", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 10, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "43b6e0c16ab4edabeba85f26b76942cd2fd314c06630777566048d3f92c4fb78", "textHash": "5927dfc145b646232aefdc50c5239d1d02d5493d0df49ac8ea2baa3376a1d1c8", "characters": 1557, "flags": [], "candidateHeadings": [ "Challenge Problems", "1.13 We can count by 9's or list the values of 9n for n = 1, 2, ..., 8; 9, 18, 27, 36, 45, 54, 63, and 72.", "Challenge Problems", "1.16 Let Scott's score be 81 + n, where n is an integer between 0 and 6 inclusive. The average of all four scores is \\(\\frac{94 + 91 + 95 + (81 + n)}{4} = \\frac{361 + n}{4} = 90 + \\frac{1 + n}{4}\\).", "1.18 We can narrow down the range by noting that" ], "candidateMarkers": [ { "id": "1.13", "label": null, "offset": 19 }, { "id": "1.14", "label": null, "offset": 127 }, { "id": "1.15", "label": null, "offset": 202 }, { "id": "1.16", "label": null, "offset": 661 }, { "id": "1.17", "label": null, "offset": 1073 }, { "id": "1.18", "label": null, "offset": 1283 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0010.png", "sha256": "a77143234a0e10cbcc07f642e5dbca9e8ec076213fc24eb1d19d96682c31df87", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 11, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "72fd155dfc82c0d7736a4043537c40273ffb95c0e60d0585579310c777d1ca10", "textHash": "c904e529c2fced125c8029f041da6afa7e86a61b05278c9c34d530f801db0112", "characters": 2496, "flags": [], "candidateHeadings": [ "1.19 First, we determine the largest possible exponent of a perfect power between 1 and 100. We do this by noting that \\(2^m < n^m\\) for any positive integer \\(n > 2\\) and any positive integer \\(m\\). This means we can determine the largest possible power of a perfect power less than 100 by determining the largest power of 2 less than 100:", "1.20 This problem is only harder than other divisor problems in that any mistake could throw our count off. The integers between 1 and 9 inclusive that are divisors of 24,516 are 1, 2, 3, 4, 6, and 9, so there are \\(\\boxed{6}\\) of them. If you counted any extra or didn't include any of these, go back and check your division.", "1.21 Yes. \\(11111 \\div 41 = 271\\), an integer, so 11111 is divisible by 41.", "1.22 Yes. \\(111111 \\div 37 = 3003\\), an integer, so 111111 is divisible by 37.", "1.23 We check each of the smallest positive integers for divisibility of 5040 until we find a positive integer that is not a divisor of 5040." ], "candidateMarkers": [ { "id": "1.19", "label": null, "offset": 170 }, { "id": "1.20", "label": null, "offset": 1536 }, { "id": "1.21", "label": null, "offset": 1864 }, { "id": "1.22", "label": null, "offset": 1941 }, { "id": "1.23", "label": null, "offset": 2021 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0011.png", "sha256": "cd609836edf4fe5c0d07e3f35de65fc845a40979f3f7c8ceeb0d4815bb67504b", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 12, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ea777e97b7a6ec53be736a811e06f128cd745c7b891a5f0585f2eb0c3a2586eb", "textHash": "af41937fa42b48a9a4c978907ce482b95f157fce807e96a93f70f9817160a81c", "characters": 963, "flags": [], "candidateHeadings": [ "Challenge Problems", "1.24 The area of a circle in terms of its diameter is \\(\\frac{d^2\\pi}{4}\\), so we are looking for the value of \\(d\\) such that" ], "candidateMarkers": [ { "id": "1.24", "label": null, "offset": 482 } ], "visualReview": { "status": "scan-compared-chapter1", "evidence": "book05-source\\answer-0012.png", "sha256": "18a3bb29c13504229620726771734861d813f62b1e7e663868156925a052dd20", "scope": "Mathematical teaching, numbered statements, all subparts, supplied solutions and diagram labels", "continuationBoundary": "Next chapter starts book PDF40; Section1.3 exercises continue onto26;1.5 exercise on33." } }, { "source": "answers-05", "pdfPage": 13, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "aff81ffa052e1dac34e93cec695ef47a60e911188e042266d5b6a88b42605255", "textHash": "29bc89bb67ced39fd63da6dcb2913e98552c8451d4bb3ce6222c76e36f733dea", "characters": 1364, "flags": [], "candidateHeadings": [ "Exercises for Section 2.2" ], "candidateMarkers": [ { "id": "2.2.1", "label": null, "offset": 94 }, { "id": "2.2.2", "label": null, "offset": 422 }, { "id": "2.2.3", "label": null, "offset": 590 }, { "id": "2.2.4", "label": null, "offset": 959 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/answer-0013.png", "sha256": "4f41614e3446738f315ba90030f2850b4f96083a43af4877df79d8722ba8a1e2", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "answers-05", "pdfPage": 14, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "75bf8c925358cdd1b20e72722ae3e022d97a099f03340ac761e2e4ec451aa7b4", "textHash": "e19469a5694bc95d8245c3799ec62afd3eb0b973664e6a0da3402880b41ecfc4", "characters": 669, "flags": [], "candidateHeadings": [ "Exercises for Section 2.3", "Exercises for Section 2.4" ], "candidateMarkers": [ { "id": "2.3.1", "label": null, "offset": 39 }, { "id": "2.3.2", "label": null, "offset": 104 }, { "id": "2.4.1", "label": null, "offset": 240 }, { "id": "2.4.2", "label": null, "offset": 467 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/answer-0014.png", "sha256": "067d161710d9522475a32daf63ff6af5058bb4007d4b6e5d71860874eb4d3d65", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "answers-05", "pdfPage": 15, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f5fdcf14b12a8d78317727c3ae570fb213543a416f8726fafdb15f036f3163fa", "textHash": "5cf4c550ae2e58b1adb2d5147977c1815598d03edee45ec5f5953931dbd538ee", "characters": 1334, "flags": [], "candidateHeadings": [ "Review Problems", "2.6 The circled numbers are prime and the rest are composite:", "2.7 In the Sieve of Eratosthenes, we cross out every positive multiple of 3 other than 3 itself (which is" ], "candidateMarkers": [ { "id": "2.6", "label": null, "offset": 49 }, { "id": "2.7", "label": null, "offset": 1226 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/answer-0015.png", "sha256": "b440369e0dfe991cf0eda4474bdb56443e65b1011e88d171ddfa716b24ab6b4b", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "answers-05", "pdfPage": 16, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "94d50c7c7565487820d38df87d34f7c2a39086ff103fa7d9f0fa60d2263bdec9", "textHash": "e0dae4a96bbe4aca2772b4865ba7b6bb585be8f0212f05b6073b9f1cd6949523", "characters": 2630, "flags": [], "candidateHeadings": [ "Challenge Problems", "2.8 Since 2 | 10, we know that 2 | 10n for any integer n. This means any multiple of 10 has 2 as a divisor and is therefore composite. There are \\(\\boxed{0}\\) prime multiples of 10.", "2.10 Let n be the number of students at class. Betty noticed that n has no divisors between 1 and itself, so n is prime. Wilma noticed that n + 2 is prime. This means we are looking for the smaller of two primes that differ by 2 that are between 30 and 50. The primes in that range are 31, 37, 41, 43, and 47. Since 41 and 43 differ by 2, n = \\(\\boxed{41}\\).", "2.11 A two-digit number whose digits are prime is made up of the digits 2, 3, 5, or 7. We check the possible numbers from the largest in this list and work backwards until we find a prime number: 77 is divisible by 7.", "2.12 A perfect cube is the product of three integers: \\(n \\cdot n \\cdot n = n^{3}\\). Unless n < 2, the perfect cube has a divisor between 1 and \\(n^{3}\\) (namely n) and is therefore composite. When n = 1, \\(n^{3} = 1\\), which is also not prime, so there are \\(\\boxed{0}\\) prime cubes.", "Challenge Problems", "2.13 The sum of the six smallest primes is \\(2 + 3 + 5 + 7 + 11 + 13 = 41\\). The seventh is 17 and the remainder when 41 is divided by 17 is \\(\\boxed{7}\\).", "2.14 A number with digits xy has value \\(10x + y\\), while the number obtained while reversing its digits, yx, has value \\(10y + x\\). Setting the sum of these two equal to 110, we get" ], "candidateMarkers": [ { "id": "2.8", "label": null, "offset": 81 }, { "id": "2.9", "label": null, "offset": 264 }, { "id": "2.10", "label": null, "offset": 1249 }, { "id": "2.11", "label": null, "offset": 1609 }, { "id": "2.12", "label": null, "offset": 1938 }, { "id": "2.13", "label": null, "offset": 2244 }, { "id": "2.14", "label": null, "offset": 2401 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/answer-0016.png", "sha256": "188785d8c8991815557ecbf66a5e4c55b75f1333a7ac9b5dbfa84593b37c374f", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "answers-05", "pdfPage": 17, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9da4c6595d79c68f6362884008040ce2411a634872fad0cabaa979a94e351067", "textHash": "b4944a23e942aab39629d287b670f513eef57311496c4f8bbd94fecad08e4b0c", "characters": 2424, "flags": [], "candidateHeadings": [ "2.15 Since \\(\\underline{523}\\) and \\(\\underline{717}\\) are both odd, their sum is even and therefore divisible by 2. There are no smaller \\(\\underline{2}\\), so it is the smallest prime divisor of the sum.", "2.16 We must find the largest prime c such that \\(a + b + c = 25\\) where a and b are also (different) \\(\\underline{1}\\)primes. One method is to check the largest possible values of c working downwards.", "2.17 All two-digit numbers whose digits sum to 9 are composite:", "2.19 So long as we can find 5 \\(\\underline{1}\\)primes between 1000 and 37² = 1369, we can check the integers after 1000 one by one to see which have no prime divisors less than 37. The Sieve of \\(\\underline{1}\\)nygard speeds up the process.", "2.20 None of the even years after 2000 are prime, and 2001 is a multiple of 3. Now we check to see if 2003 is \\(\\underline{1}\\)prime." ], "candidateMarkers": [ { "id": "2.15", "label": null, "offset": 415 }, { "id": "2.16", "label": null, "offset": 620 }, { "id": "2.17", "label": null, "offset": 1183 }, { "id": "2.18", "label": null, "offset": 1424 }, { "id": "2.19", "label": null, "offset": 1554 }, { "id": "2.20", "label": null, "offset": 2005 } ], "visualReview": { "status": "scan-compared-chapter2", "evidence": "book05-source/answer-0017.png", "sha256": "606664d1b1987b1884f710d219f83893545765539e8428aa51f73a61b45ae65c", "continuationBoundary": "Next chapter54; Section2.3 ends midway46, after continuation sieve explanation." } }, { "source": "answers-05", "pdfPage": 18, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "bd74e7d65db39baa8e44248d9144f1241a512dd764a560dff36de2a4c0d69b10", "textHash": "afff397b3a6b42e6cebac110dba3d74b62c5e13ce07a318b016ea270aaac446f", "characters": 1736, "flags": [], "candidateHeadings": [ "Exercises for Section 3.2" ], "candidateMarkers": [ { "id": "3.2.1", "label": null, "offset": 59 }, { "id": "3.2.2", "label": null, "offset": 337 }, { "id": "3.2.3", "label": null, "offset": 618 }, { "id": "3.2.4", "label": null, "offset": 852 }, { "id": "3.2.5", "label": null, "offset": 1012 }, { "id": "3.2.6", "label": null, "offset": 1413 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0018.png", "sha256": "103efa895b6ad9ff04a7265ec3d0374b0b49b5243ccc3af8563d13ebb5989ec3", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 19, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "df4d9a8fb39cf2c824a808991472fa0f4f30ae0bacebff9a9b6dc58c5ed35d96", "textHash": "1782d4bd01e40558106ab3dd35b4e4352c4d7743ed2d6c9c4f64ba4330893621", "characters": 1242, "flags": [], "candidateHeadings": [ "Exercises for Section 3.3", "Exercises for Section 3.4", "Exercises for Section 3.5" ], "candidateMarkers": [ { "id": "3.3.1", "label": null, "offset": 61 }, { "id": "3.3.2", "label": null, "offset": 238 }, { "id": "3.4.1", "label": null, "offset": 422 }, { "id": "3.4.2", "label": null, "offset": 582 }, { "id": "3.5.1", "label": null, "offset": 862 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0019.png", "sha256": "71d235036013586ba80bae2d502c17412b5d1381de6cf968f6c480d9cabaab5f", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 20, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "55d69bc0936ed5d4a449486360617b996d648e8c3ac81ce53e8e5762d32dd917", "textHash": "067cbecef81752aa43681e1dd16e2b8a816543a17298b8c7006e13e16005115d", "characters": 2213, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "3.5.2", "label": null, "offset": 169 }, { "id": "3.5.3", "label": null, "offset": 884 }, { "id": "3.5.4", "label": null, "offset": 1357 }, { "id": "3.5.5", "label": null, "offset": 1805 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0020.png", "sha256": "0954cbef45a9e23d0997a8358f7bfdb485939fdb28ba749136cde5a4f3d9c3c4", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 21, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "04c3d30e4e4271cae763a79f10e3571cf89faa5dae302048520885184c04b707", "textHash": "881b8037282864890fbe9605ad91a3eb7cb374456046eaafa535ab902a8064ed", "characters": 1542, "flags": [], "candidateHeadings": [ "Exercises for Section 3.6" ], "candidateMarkers": [ { "id": "3.5.6", "label": null, "offset": 727 }, { "id": "3.6.1", "label": null, "offset": 1191 }, { "id": "3.6.2", "label": null, "offset": 1386 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0021.png", "sha256": "c721bc0afc6d4c034cd883c6b7c16cc44f2348695c4b1646044020c90c2403bc", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 22, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "494ebfafc16916d9d05c3dd81c6bba80583548ad2a09bfb66e770d3d239088d2", "textHash": "75aa884a7ca7665a3cf89d743338b84704854bc82cb1d1f222ee66d412061792", "characters": 1427, "flags": [], "candidateHeadings": [ "Exercises for Section 3.7" ], "candidateMarkers": [ { "id": "3.6.3", "label": null, "offset": 12 }, { "id": "3.6.4", "label": null, "offset": 229 }, { "id": "3.6.5", "label": null, "offset": 449 }, { "id": "3.6.6", "label": null, "offset": 723 }, { "id": "3.7.1", "label": null, "offset": 1026 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0022.png", "sha256": "5c208d0fb698f60f56aec17b9391d482a97dfb274d716dedf2385fcdeda47306", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 23, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "0964e58d536aa52425e1413bee1fe73b297d88feb8b7fcebf7aa6a737b0a9690", "textHash": "b4e313b6f11ad8737dce63513eb09f3b9c972570dc08593ff0b467d5c9a8e14c", "characters": 1499, "flags": [], "candidateHeadings": [ "Review Problems", "3.26 It's probably easier to first find the LCM of two of the integers. The LCM of 12 and 18 is 36. Every common multiple of 12 and 18 will be a multiple of their LCM, so the LCM of all three integers must be a multiple of 36. The LCM of all three integers is the LCM of 36 and 30, which is 180.", "3.27 Let x be this smallest number of marbles that can be broken up into bags of 18 or bags of 42. This means x must be a multiple of both 18 and 42. The smallest such number is lcm[18, 42] = 126, so x = 126.", "3.28 gcd(6432, 132) = gcd(96, 132) = gcd(96, 36) = gcd(24, 36) = gcd(24, 12) = gcd(0, 12) = 12. Decreasing this number by 8, the result is 12 - 8 = 4." ], "candidateMarkers": [ { "id": "3.23", "label": null, "offset": 50 }, { "id": "3.24", "label": null, "offset": 638 }, { "id": "3.25", "label": null, "offset": 705 }, { "id": "3.26", "label": null, "offset": 841 }, { "id": "3.27", "label": null, "offset": 1137 }, { "id": "3.28", "label": null, "offset": 1346 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0023.png", "sha256": "1abbbafcb1a0dfda619df6f52b27fe4f84bf8e27d381506c657d866d3bca9d1a", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 24, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a44804db74f0e4960ba4f236c3a9f1ad61b2f94bede218c520d8b4d98760d86f", "textHash": "f361855394d7921ac91d0f1172ec0826dc1dbfd231ce7a84e2703a04685e0a89", "characters": 2513, "flags": [], "candidateHeadings": [ "Challenge Problems", "3.29 An integer that is divisible by 7 can be written in the form 7n for some integer n. Our goal is to count the number of integers such that", "Challenge Problems", "3.30 The number of cookies in each package must be a divisor of each of 252, 105, and 168. Since gcd(252, 105) = 21, we check to see if 21 is also a divisor of 168. It is, so \\(\\underline{21}\\) is the greatest possible number of cookies in each package.", "3.31 There are many ways to create each multiple of 5 out of combinations of 25's and 15's, but here are simple ones for the 10 smallest positive multiples of 5:", "3.32 Of the 25 primes less than 100, only 1 is even. The other 24 are odd and their sum is therefore even. Adding the even prime, 2, the sum of all 25 primes is \\(\\underline{even}\\).", "3.33 There are 20 positive multiples of 5 less than 101. There are 14 positive multiples of 7 less than 101. However, there are 2 positive common multiples of 5 and 7 less than 101. This means there are 18 multiples of 5 that aren't multiples of 7 and 12 multiples of 7 that are not multiples of 5 for a total of \\(18 + 12 = \\underline{30}\\).", "3.34 The key to solving this problem without using prime factorization (which we'll get to later in the book) is using variables to express every piece of information and relating all quantities in terms of multiplication by integers." ], "candidateMarkers": [ { "id": "3.29", "label": null, "offset": 19 }, { "id": "3.30", "label": null, "offset": 403 }, { "id": "3.31", "label": null, "offset": 658 }, { "id": "3.32", "label": null, "offset": 1320 }, { "id": "3.33", "label": null, "offset": 1711 }, { "id": "3.34", "label": null, "offset": 2274 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0024.png", "sha256": "a53a874d4eb6786b5ca7e35535b068b457a23f1afaf92c9624a9261a63788102", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 25, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a8460531858915330eabb268060950d4c2746d9a717d33e3c10737eddc4053d1", "textHash": "605904109eb29e10ad8d177161db0167e2da8922b5eec9ab245a853a347f8525", "characters": 2200, "flags": [], "candidateHeadings": [ "3.35 Using the kind of algebra we use in the Division Theorem, we look for a four-digit number a and a one-digit number b such that", "3.36 There are equally many a’s, b’s, c’s, and d’s on the left-hand sides of the equations, so we add them to get a “balanced” equation:" ], "candidateMarkers": [ { "id": "3.35", "label": null, "offset": 786 }, { "id": "3.36", "label": null, "offset": 1268 } ], "visualReview": { "status": "scan-compared-chapter3", "evidence": "book05-source/answer-0025.png", "sha256": "a660c7c412d3cef6c3dcc8c0f6ca57d0abadf642cff832888968581ffafba6cd", "nextChapterHeadingPDF": 78 } }, { "source": "answers-05", "pdfPage": 26, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "54c98623b54821dfc0eba42aadd5ba183b37fcf6cccbc3e3412387617d6c8204", "textHash": "df42acf241c957671b3051fc4869236bc6ecb28994bb075bec51100e2dc7d260", "characters": 6389, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [ "Exercises for Section 4.2" ], "candidateMarkers": [ { "id": "4.2.1", "label": null, "offset": 54 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0026.png", "sha256": "feed9cab1e85ce1e064205672445b7ab82b0e590ac007bb92c2b6fb090b076d9", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 27, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "0672e58afaee91ca079cd69ba7f1674c6156ca7bb40e1da833d45734538647b8", "textHash": "448ade98c24abef8ed7539fcf43c75d78c21196ebc93afb5daddf7a40043935c", "characters": 1855, "flags": [], "candidateHeadings": [ "Exercises for Section 4.4" ], "candidateMarkers": [ { "id": "4.3.2", "label": null, "offset": 31 }, { "id": "4.3.3", "label": null, "offset": 436 }, { "id": "4.3.4", "label": null, "offset": 732 }, { "id": "4.4.1", "label": null, "offset": 1018 }, { "id": "4.4.2", "label": null, "offset": 1441 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0027.png", "sha256": "ddc377bbf5538ca35b214b104f0d12758967735118f6fffdc72e7eb1150cbea6", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 28, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "24edb45895ec9b49260b90c00827e347439389acd1e9031caffb4f9123537a8a", "textHash": "fe576bbfa0fa532debede87252ba47af100520838a1615c70e4921a1f86d8c2c", "characters": 1603, "flags": [], "candidateHeadings": [ "Exercises for Section 4.5", "Exercises for Section 4.6" ], "candidateMarkers": [ { "id": "4.5.1", "label": null, "offset": 39 }, { "id": "4.6.1", "label": null, "offset": 964 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0028.png", "sha256": "ca178fdbd787d5d7a3c76b6fa06e7b7fc0f28b890abde9290c3a3aab4803f4ee", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 29, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "235c654c5c1d106a0b84ab080aabb2edfa354e7814eff58bed88e6459c72f523", "textHash": "4cbe332ad798b8e3199a83ace487c7e9b6fc9c15e37861dfacf2d127fbcf5600", "characters": 1670, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "4.6.2", "label": null, "offset": 30 }, { "id": "4.6.3", "label": null, "offset": 266 }, { "id": "4.6.4", "label": null, "offset": 715 }, { "id": "4.6.5", "label": null, "offset": 1227 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0029.png", "sha256": "fef4f4a69be8ac6ff52525eb5dec76d28568ecb50156eac57c5dee9111c92daf", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 30, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "e36bae915808276989ba433651d25ec1239dc3cf75e4fb974bee0f5ed06f0d5e", "textHash": "5a6c9974a0876d9d83159347de5397b5b30d75401a979be6aa508f31dc784ad1", "characters": 1387, "flags": [], "candidateHeadings": [ "Exercises for Section 4.7", "Review Problems" ], "candidateMarkers": [ { "id": "4.7.1", "label": null, "offset": 39 }, { "id": "4.7.2", "label": null, "offset": 365 }, { "id": "4.7.3", "label": null, "offset": 420 }, { "id": "4.7.4", "label": null, "offset": 510 }, { "id": "4.24", "label": null, "offset": 646 }, { "id": "4.25", "label": null, "offset": 788 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0030.png", "sha256": "f53784138e35de9ab314a09979576e178194530b7e320294984d45aa253ee025", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 31, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "6f351188b921259d2f3a016bb6bd7903b9f6132ea73612dfa6a4ffc42bb864cf", "textHash": "5ac076395b3d24490fadf84c7f19ed3e6555a4cc67691a9e308ed5578c30df72", "characters": 2327, "flags": [], "candidateHeadings": [ "4.26 Common multiples of 54 and 99 are multiples of \\(\\mathrm{lcm}[54, 99] = 594\\), so they can be expressed in the form \\(594n\\). The 8 smallest positive common multiples of 54 and 99 are", "4.27 We want to find the smallest four-digit number that is a multiple of \\(\\mathrm{lcm}[2, 3, 5, 7] = 210\\), so we need to find the smallest value of \\(n\\) such that", "4.28 The positive common divisors of 84 and 132 are the positive divisors of \\(\\gcd(84, 132) = 12 = 2^2 \\cdot 3^1\\). These divisors are formed by multiplying the powers of 2 from \\(2^0\\) up to \\(2^2\\) by the powers of 3 from \\(3^0\\) up to \\(3^1\\):", "4.29 The positive common divisors of 162 and 540 are the positive divisors of \\(\\gcd(162, 540) = 54 = 2^1 \\cdot 3^2\\). These divisors are formed by multiplying the powers of 2 from \\(2^0\\) up to \\(2^1\\) by the powers of 3 from \\(3^0\\) up to \\(3^3\\):" ], "candidateMarkers": [ { "id": "4.26", "label": null, "offset": 323 }, { "id": "4.27", "label": null, "offset": 791 }, { "id": "4.28", "label": null, "offset": 1139 }, { "id": "4.29", "label": null, "offset": 1568 }, { "id": "4.30", "label": null, "offset": 2061 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0031.png", "sha256": "9f2bf9ea6669bd7dd39f7812473373bc32bc6e5f65c64aec8472efa03bbad747", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 32, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2a07005bd8a4f646d8e83c671b413f938f73fe6dacf854c989c17e7f8506ebe1", "textHash": "baa35339dd2f541c63f2d529806d6504e12afe2f52f39c9dd2e0d265e64b30b8", "characters": 2214, "flags": [], "candidateHeadings": [ "Review Problems", "4.33 Positive multiples of 6 = 2¹ · 3¹ that are perfect squares are multiples of 2² · 3² = 36 that can be written in the form 36n² for each positive integer n. Positive multiples of 6 that are perfect cubes are multiples of 2³ · 3³ = 216 that can be written in the form 216n³ for each positive integer n.", "4.34 The product of two positive integers is equal to the product of their GCD and LCM, so", "4.35 Since the product of two integers is equal to the product of their GCD and LCM, we have the equation" ], "candidateMarkers": [ { "id": "4.31", "label": null, "offset": 460 }, { "id": "4.32", "label": null, "offset": 901 }, { "id": "4.33", "label": null, "offset": 1354 }, { "id": "4.34", "label": null, "offset": 1931 }, { "id": "4.35", "label": null, "offset": 2079 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0032.png", "sha256": "d9a21174ffabce4ef176e7c067c063a41623edcb942411c6c6ef1d067bda70e9", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 33, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "1d19afbb542c50c8e87b99ca51b090c7500256d1ecb975abd4e5d80ae05dfdb2", "textHash": "6fcafb5084571fbae1503218fd162e9a8f67ee598b63bbb97233b0724baa4894", "characters": 2629, "flags": [], "candidateHeadings": [ "Challenge Problems", "4.39 First, none of the first five friends could have been incorrect. If one of them were, and their number was n, then friend 2n would also be incorrect, but the incorrect friends were consecutively numbered, so they would have to be friends 1 and 2 and friend 1 cannot possibly have been incorrect.", "4.40 Since 2, 3, and 5 divide 30N but not 7, they do not divide 30N + 7. Similarly, 7 only divides 30N + 7 if 7 divides 30N, which means N must be a multiple of 7 for 7 to divide it. Since no number less than 11 divides 30N + 7 while N < 7, we only need to check when \\(30N + 7 \\geq 11^2\\). When N = 4, 30N + 7 = 127 is prime. When N = 5, 30N + 7 = 157 is prime. However, when N = \\(\\boxed{6}\\), 30N + 7 = 187 = 11 \\cdot 17 is composite." ], "candidateMarkers": [ { "id": "4.36", "label": null, "offset": 410 }, { "id": "4.37", "label": null, "offset": 563 }, { "id": "4.38", "label": null, "offset": 708 }, { "id": "4.39", "label": null, "offset": 1269 }, { "id": "4.40", "label": null, "offset": 2187 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0033.png", "sha256": "5d9676d41d9f6bc74b7398b03ee4453aa2c1e8798a6c4d6a0fb6392fa7142b0d", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 34, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "8dca809a8df8ab6cea8028b72ba10b87d3bcfe866ab5ee603fe80188bb8dc9d2", "textHash": "55d8dcff952cad641fbef497acab99985b56863d2984c7934c9ddc4d8b8bc3a2", "characters": 2650, "flags": [], "candidateHeadings": [ "Challenge Problems", "4.41 Since perfect squares that are perfect cubes have exponents in their prime factorizations that are both multiples of 2 and multiples of 3, all the exponents must be multiples of 6. Common multiples of 18 and 30 are multiples of \\(1cm\\) [18, 30] = 90 = 2¹ · 3² · 5¹. A multiple of 90 that is both a square and a cube must be in the form", "4.42 None of the integers is 1 because we could always remove it to reduce the sum while keeping the product the same. Now, note that the sum of two integers both greater than 1 is never greater than their product.", "4.43 Any common multiple of 14, 26, and 66 must be a multiple of their LCM. No common divisor of these multiples can be larger than this LCM (since it must be a divisor of the LCM). Therefore, the GCD of the common multiples of 14, 26, and 66 is the LCM of 14, 26, and 66.", "4.44 In general, when a proper divisor d of an integer n is large, the value of \\(\\frac{n}{a}\\), which must itself be a proper divisor of n, is small. Since 1999, 2000, and 2001 are very close in value, we are really looking for the one that has the smallest divisor greater than 1. The smallest possible divisor greater than 1 is 2, and since \\(\\frac{2000}{2} = 1000\\) is a proper divisor of \\(\\frac{2000}{2}\\), we have our answer." ], "candidateMarkers": [ { "id": "4.41", "label": null, "offset": 19 }, { "id": "4.42", "label": null, "offset": 769 }, { "id": "4.43", "label": null, "offset": 1595 }, { "id": "4.44", "label": null, "offset": 2009 }, { "id": "4.45", "label": null, "offset": 2443 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0034.png", "sha256": "d6cc3393cc4a24da84aaf22db776be65532b964f02adf03640c773ad64567894", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 35, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "5fb23fba66282650af1371b024bd35ee0247b221a242bc2e722dffeb45e5fa77", "textHash": "f9d05b3c6bcb2f8b79858ca90e97e7044b83ba9614f1b6331dfda80643ada87d", "characters": 2543, "flags": [], "candidateHeadings": [ "4.46 For a positive integer n, we know that \\(8n = 2^3 \\cdot n\\) has 3 more powers of 2 in its prime factorization. This means that there is an even power of 2 in the prime factorization of one of n and 8n and an odd power of 2 in the other. Since all the exponents in the prime factorization of a perfect square must be even, at least one of n and 8n be must non-square.", "4.47 The sum of the reciprocals is a fraction with denominator 60 in lowest terms. This means that 60 is a divisor of the LCM of the three consecutive integers.", "4.48 The number of stamps Jenna puts on each page must divide the number of stamps she puts into each book, so the largest possible number of stamps she puts on each page is \\(\\gcd(840, 1008) = \\boxed{168}\\).", "4.49 We use the formula for the sum of the n smallest positive integers, \\(\\frac{n(n+1)}{2}\\).", "4.50 Let n be an integer that leaves a remainder of 1 when divided by each of 2, 3, ..., 8, and 9. That means n - 1 is a common multiple of 2, 3, ..., 8, and 9 and therefore a multiple of" ], "candidateMarkers": [ { "id": "4.46", "label": null, "offset": 315 }, { "id": "4.47", "label": null, "offset": 688 }, { "id": "4.48", "label": null, "offset": 1586 }, { "id": "4.49", "label": null, "offset": 1796 }, { "id": "4.50", "label": null, "offset": 2021 }, { "id": "4.51", "label": null, "offset": 2500 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0035.png", "sha256": "d4f8413ff422d6ac321319cfce40a91a421cf8bfab9808be40de84ac8d5c9b24", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 36, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "86efdf7822f26fd03b278490c9e12e53bde47a062cf88a29a7d04fba9a302a60", "textHash": "74465d9f09d4f39512c5bd5eede586a2ac856a4c2c392cb8f19f9eff42e9568c", "characters": 1948, "flags": [], "candidateHeadings": [ "Challenge Problems", "4.52 Jayne erases every integer in the list that includes any prime greater than 2 in its prime factorization. This means the only integers left are the powers of 2." ], "candidateMarkers": [ { "id": "4.52", "label": null, "offset": 416 }, { "id": "4.53", "label": null, "offset": 671 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0036.png", "sha256": "17864398d24d974faddf58060ece6efb5b5fd65f83e08a548d5ced9b63a54f15", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 37, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "93826e72502808f83edc3fde35f52fc50e2ad126ed02652498d9431d7687dd5f", "textHash": "328ca1f5b73d5b22c37cbd49851772374e01a9bf875c9be32f91b14d63951c32", "characters": 2112, "flags": [], "candidateHeadings": [ "4.54 Matthew and Alex count the same number at every common multiple of 6 and 4. Every common multiple of 6 and 4 is a multiple of \\(\\lfloor \\text{cm}[6, 4] \\rfloor = 12\\). Positive multiples of 12 can be written as \\(\\lfloor 12n \\rfloor\\) for positive integers n. These are no greater than 400:", "4.55 Let's evaluate the three pieces of information given by the problem:" ], "candidateMarkers": [ { "id": "4.54", "label": null, "offset": 335 }, { "id": "4.55", "label": null, "offset": 739 }, { "id": "4.56", "label": null, "offset": 1601 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0037.png", "sha256": "ebf84a7d57a437998e04344849882c55580ee668460055ec8fecd71768989d3d", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 38, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "6b2ff83ab7ce396793ad4d777afa1bf0f2b28a050054655670ac517538f243fc", "textHash": "9bedf316d3a22478ad574e6176cc9f181423b7e928749134c4b2ef5ffea34c5a", "characters": 1217, "flags": [], "candidateHeadings": [ "Challenge Problems", "4.57 Since the numbers are not all different, at least two of them are the same, so we can let the numbers be x, x and y. Since \\(x^2 y = n^2\\) is a perfect square," ], "candidateMarkers": [ { "id": "4.57", "label": null, "offset": 583 } ], "visualReview": { "status": "scan-compared-chapter4", "evidence": "book05-source/answer-0038.png", "sha256": "b7921dd3706bf06598d4c57cf02e96d07e4627b9efba52984a6246b875711fee", "nextChapterHeadingPDF": 106 } }, { "source": "answers-05", "pdfPage": 39, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "b6ac4a69073d7987105eb3a761b4309a44089082e49cd763a88bec32ee3c941f", "textHash": "9974fe188c8316aa111c061d1267c3e363fe108cb9beadb3b7e4d1b2464c27e2", "characters": 1069, "flags": [], "candidateHeadings": [ "Exercises for Section 5.2" ], "candidateMarkers": [ { "id": "5.2.1", "label": null, "offset": 80 }, { "id": "5.2.2", "label": null, "offset": 735 }, { "id": "5.2.3", "label": null, "offset": 808 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0039.png", "sha256": "e3fec3f1c46fb95d81365ce655757269081c77c7f11db9efb2167df4f2d4a2cd", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 40, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "58dc39cc517790973cf150ae6a8e144ba2e139f9496ea30a338ccec5002dacc5", "textHash": "a8fb62f84d375ba13e3500bd1bef5546cc87551ca136c76505692b228ec150b6", "characters": 2293, "flags": [], "candidateHeadings": [ "Exercises for Section 5.3" ], "candidateMarkers": [ { "id": "5.3.1", "label": null, "offset": 39 }, { "id": "5.3.2", "label": null, "offset": 501 }, { "id": "5.3.3", "label": null, "offset": 865 }, { "id": "5.3.4", "label": null, "offset": 1196 }, { "id": "5.3.5", "label": null, "offset": 1310 }, { "id": "5.3.6", "label": null, "offset": 1536 }, { "id": "5.3.7", "label": null, "offset": 1893 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0040.png", "sha256": "3ac23c340f96bb73806b0d9f984938f63ab16a79274546f41df93acf450133da", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 41, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "5afa45e099bc9370c415c6ea26db899aae25a1bde0e5d9db6a3e1e15814f056f", "textHash": "e236f8a54a8793c6a5672b8c695121d6aa67b9694009768a8b0288170a781ecb", "characters": 2770, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.3.8", "label": null, "offset": 268 }, { "id": "5.3.9", "label": null, "offset": 1127 }, { "id": "5.3.10", "label": null, "offset": 1577 }, { "id": "5.3.11", "label": null, "offset": 2495 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0041.png", "sha256": "e86a96a5c4f96df24e448940ff240ff027d862fdf59f541a7b8a23725b6a0e1e", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 42, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9190d6822fe93d702b5c444187823c66da55f75d75392557825be0ecc4e588dd", "textHash": "c495cbc28fd9a007ce664ab89d10fe96cb55f55153715005ed87ef4415cd74d5", "characters": 2703, "flags": [], "candidateHeadings": [ "Exercises for Section 5.4" ], "candidateMarkers": [ { "id": "5.4.1", "label": null, "offset": 39 }, { "id": "5.4.2", "label": null, "offset": 189 }, { "id": "5.4.3", "label": null, "offset": 345 }, { "id": "5.4.4", "label": null, "offset": 486 }, { "id": "5.4.5", "label": null, "offset": 776 }, { "id": "5.4.6", "label": null, "offset": 1101 }, { "id": "5.4.7", "label": null, "offset": 1724 }, { "id": "5.4.8", "label": null, "offset": 2160 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0042.png", "sha256": "948f683455f1ec7875ea9add354b9cd47619a9be4e0a0d3a5998ffb3cc0eda43", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 43, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "4d14cd0ec7272ac86ddec499bd283a6f2d34599651ea290432f5f2d345fc1a09", "textHash": "3237d3c84f7136bd7da19b00686f01e3da46249547c85ddfe6b767964cea4af6", "characters": 2644, "flags": [], "candidateHeadings": [ "Review Problems", "5.15 Timothy's quotient is an integer if and only if Joseph's number is a divisor of 1000. Our goal is to count the positive divisors of \\(1000 = 2^{3} \\cdot 5^{3}\\), so \\(t(1000) = (3 + 1)(3 + 1) = 16\\) is the number of integers that would make Timothy's quotient an integer.", "5.17 The perfect square divisors of 3240 are in the form \\(2^{2a} \\cdot 3^{2b} \\cdot 5^{2c}\\), where \\(0 \\leq a \\leq 1\\), \\(0 \\leq b \\leq 2\\), and \\(c = 0\\). This gives \\(3 \\cdot 2 \\cdot 1 = 6\\) choices for the exponents to the primes that make up the prime factorizations of the perfect square divisors of 3240." ], "candidateMarkers": [ { "id": "5.13", "label": null, "offset": 43 }, { "id": "5.14", "label": null, "offset": 621 }, { "id": "5.15", "label": null, "offset": 817 }, { "id": "5.16", "label": null, "offset": 1094 }, { "id": "5.17", "label": null, "offset": 1473 }, { "id": "5.18", "label": null, "offset": 1786 }, { "id": "5.19", "label": null, "offset": 2279 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0043.png", "sha256": "288220bb3d133666a20c4c7a68685e86bf6cb6437ef9cfa92fe1cdec0550ec7d", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 44, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "8ef119efb437b4762bcd9f14dc5272b49719d12b81d1c727e66623f350984f42", "textHash": "88d72ba071c8eb05f0a9f74a04a53d3dc9c0fcc152097e011b98ab37a1e943c7", "characters": 2759, "flags": [], "candidateHeadings": [ "Challenge Problems", "Challenge Problems", "5.21 Since \\(m\\) and \\(n\\) are relatively prime, the primes in their prime factorizations are entirely distinct. This means that the exponents in the prime factorization of \\(mn\\) are simply all those in the prime factorizations of \\(m\\) and \\(n\\). Since we multiply 1 more than each of those exponents to count divisors, we have \\(t(mn) = t(m)t(n) = 12 \\cdot 10 = 120\\) positive divisors of \\(mn\\). We multiply these divisors by noting that together they make 120/2 = 60 pairs of divisors with products of \\(mn\\). The product of all the divisors is the product of these pairs, which is \\(\\boxed{mn}\\)\\(\\boxed{60}\\).", "5.22 For a positive integer \\(n\\), \\(t(n)\\) is a product of choices for exponents in the prime factorizations of the divisors of \\(n\\). So, when \\(t(n) = p\\) for some prime number \\(p\\), there must be only one number in that product. This means there is only 1 exponent of \\(\\boxed{1}\\) prime in the prime factorization of \\(n\\).", "5.23 If \\(t(n) = 5\\) for some integer \\(n\\), then \\(n = p^4\\) for some prime number \\(p\\). In fact, we must be looking for the fourth powers of the three smallest primes. Their sum is", "5.25 Let \\(n = p^6 \\cdot q^b\\) where \\(p\\) and \\(q\\) are prime. We are given that \\(t(n) = (a + 1)(b + 1) = 9\\). Since both \\(a + 1\\) and \\(b + 1\\) are divisors of 9 greater than 1, they must both be 3, so \\(a = b = 2\\).", "5.26 Since \\(n\\) can be written as the product of 2 factors in exactly 3 different ways excluding \\(n \\cdot 1\\), there are 4 ways in which \\(n\\) can be written as a product of divisors. This means that \\(t(n) = 8\\) when \\(n\\) is not a perfect square and \\(t(n) = 7\\) when \\(n\\) is a perfect square (where two of the factors whose product is \\(n\\) are the same)." ], "candidateMarkers": [ { "id": "5.20", "label": null, "offset": 39 }, { "id": "5.21", "label": null, "offset": 144 }, { "id": "5.22", "label": null, "offset": 762 }, { "id": "5.23", "label": null, "offset": 1093 }, { "id": "5.24", "label": null, "offset": 1330 }, { "id": "5.25", "label": null, "offset": 2079 }, { "id": "5.26", "label": null, "offset": 2393 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0044.png", "sha256": "d589ff16abb22407192f0043149a715692e7ae8306b889c103f5372c27cb24c0", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 45, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "6796e3ac6ac2f3851c8a402529be659a0aef9e7073b4643994ca2cdc02e6cfd1", "textHash": "879079329f6e43488a3fc247fdd310293c0e576c6b3d12830c1f99ff70b2be64", "characters": 2571, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "5.27", "label": null, "offset": 903 }, { "id": "5.28", "label": null, "offset": 1887 }, { "id": "5.29", "label": null, "offset": 2086 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0045.png", "sha256": "dcadd37bfef1fd1b471a1324c461d75f9bdf5ddd1b0a59021afda9e9ba7586b1", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 46, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "674a5f47d61f5a00576889031e2ba2fe4760194315a2de76f77abb3d18a7418a", "textHash": "c4102f06512627c83a7af5dd63f20712e3fc81e4d82ba2b32cf8c9912c47291b", "characters": 2826, "flags": [], "candidateHeadings": [ "Challenge Problems" ], "candidateMarkers": [ { "id": "5.30", "label": null, "offset": 1826 }, { "id": "5.31", "label": null, "offset": 2302 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0046.png", "sha256": "82fc10e0bc9cdc9f32bb05d31d520bbfeec8f34417756bc70aa44e2160ae8226", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 47, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "0ed4ea85740a6aefb97d678f45b47b29f9c5cfcfd487b180dabf77929d189f37", "textHash": "b74f8aff6c42e149132dcf6437c6d9e1edfd1b8d7398000b3b0703c44363ee0e", "characters": 1907, "flags": [], "candidateHeadings": [ "5.38 First, note that 6 perfect square divisors come from 6 choices for the possible even exponents of divisors. This means either there is 1 exponent (with 6 choices) or two exponents (with 2 and 3 choices). Since we are looking for the smallest integer n with 6 perfect square divisors, we can simply test these cases with the smallest possible primes:" ], "candidateMarkers": [ { "id": "5.32", "label": null, "offset": 599 }, { "id": "5.38", "label": null, "offset": 1147 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0047.png", "sha256": "bfb0f1de18af2c5c20214b6ec741bc2e3a3e8770d47ce78f7818752b726f6371", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 48, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "843635bb080e234ba370bf285abe61fa6bfc18b2b92fddad39458eef4fd46c41", "textHash": "f2121dab6e534ac8c1b2be020e76986ee9bc6d32b539418b4b807f06a06ac54f", "characters": 2369, "flags": [], "candidateHeadings": [ "Challenge Problems", "5.35 The number of positive divisors of an integer is the product of the possible exponents of each prime in the integer's prime factorization, where each possible number of choices is greater than 1. Since \\(30 = 2^1 \\cdot 3^1 \\cdot 5^1\\), we can express 30 as the product of at most 3 positive integers greater than 1. However, we must still confirm that this is possible.", "5.36 We are looking for integers with 3 proper divisors, and therefore 4 total divisors. Such numbers either have 4 possible choices for the exponent to one prime in their prime factorizations, or 2 choices for each of 2 primes. Their prime factorizations must be of one of the two forms \\(p^3\\) or \\(p^1 \\cdot q^1\\)." ], "candidateMarkers": [ { "id": "5.34", "label": null, "offset": 19 }, { "id": "5.35", "label": null, "offset": 442 }, { "id": "5.36", "label": null, "offset": 1306 } ], "visualReview": { "status": "scan-compared-chapter5", "evidence": "book05-source/answer-0048.png", "sha256": "7c0a7c0aff3deffb310cb407be603c48eb731506760dbd37b5b96e3a8240d69b", "nextChapterHeadingPDF": 124 } }, { "source": "answers-05", "pdfPage": 49, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "bb10517a0aefb9a1cc9f62e611cca13a5579392eebaa0ff0668011009f086061", "textHash": "119d77c0e431de06b54310eaa4cc5cf473bdafc3b512f28a182ba549d63091cc", "characters": 1504, "flags": [], "candidateHeadings": [ "Exercises for Section 6.2", "Exercises for Section 6.3" ], "candidateMarkers": [ { "id": "6.2.1", "label": null, "offset": 79 }, { "id": "6.2.2", "label": null, "offset": 268 }, { "id": "6.3.1", "label": null, "offset": 751 }, { "id": "6.3.2", "label": null, "offset": 948 }, { "id": "6.3.3", "label": null, "offset": 1352 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0049.png", "sha256": "04ecfcd237128a4be7619f2da555480ff5a2201ed1b93d1a4a2a916ba9b7fca0", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 50, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "03faf74d4531633993b28fec964c8fabf19a7516af88150a50831ca2a16c09a0", "textHash": "aad0ac1db680fa043f3e58c3cd768ea875feba3ab98834d379aaa3d8b6af634c", "characters": 1730, "flags": [], "candidateHeadings": [ "Exercises for Section 6.4" ], "candidateMarkers": [ { "id": "6.3.4", "label": null, "offset": 238 }, { "id": "6.3.5", "label": null, "offset": 493 }, { "id": "6.3.6", "label": null, "offset": 736 }, { "id": "6.3.7", "label": null, "offset": 1091 }, { "id": "6.3.8", "label": null, "offset": 1256 }, { "id": "6.4.1", "label": null, "offset": 1637 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0050.png", "sha256": "2ad7ee3c046c1cc9c65289234c28122b4ac90707f762946abdcc0abce37b317b", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 51, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "3a840ff8da343c31597e20feee01c30b4d161c4af63be2e970c91cbe02b6b6cb", "textHash": "1c276591c2e32b82b24960dd81ba7004a9ebfccedee39b3c47387cc90d92aeb7", "characters": 2029, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "6.4.2", "label": null, "offset": 625 }, { "id": "6.4.3", "label": null, "offset": 901 }, { "id": "6.4.4", "label": null, "offset": 1614 }, { "id": "6.4.5", "label": null, "offset": 1806 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0051.png", "sha256": "a3f182d7ffc7cf4d2169bd168cf172b49b5c88f48d0826b415d66e99871934a7", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 52, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "444f6cd908a4f407ff3898d9f642a14496ec0681df29e9d70da9b0f04bb79058", "textHash": "a13bee7c49615cec379556f820c9ce4bfc417b381f447c5f4c1aa0347d502acf", "characters": 2637, "flags": [], "candidateHeadings": [ "Exercises for Section 6.5", "Review Problems", "6.12 Since \\( 17^2 = 289 \\), any composite number less than 289 must be divisible by a prime number less than \\( \\sqrt{289} = 17 \\). The largest prime that is less than 17 is 13, so we use the Sieve of Eratosthenes on primes up to 13, and we see that \\(\\boxed{227}\\) and \\(\\boxed{229}\\) are prime, and that \\(\\boxed{239}\\) and \\(\\boxed{241}\\) are prime." ], "candidateMarkers": [ { "id": "6.5.1", "label": null, "offset": 178 }, { "id": "6.5.2", "label": null, "offset": 524 }, { "id": "6.5.3", "label": null, "offset": 825 }, { "id": "6.5.4", "label": null, "offset": 1337 }, { "id": "6.5.5", "label": null, "offset": 1570 }, { "id": "6.12", "label": null, "offset": 2279 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0052.png", "sha256": "f75bc442536794d67d1739eb5069f1c44941d416f46485a219f1113e1573c36a", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 53, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "7e59f9b59bec6d680a08138e327dbaf2708756fb5913377429733eccbdcd83ff", "textHash": "4adc72990616d6db22da40b7a63bd55d0e45dbea80f405b72873be5e4c5269d3", "characters": 1915, "flags": [], "candidateHeadings": [ "6.13 The only prime numbers that divide 30! are less than or equal to 30. So 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are primes that divide 30!, and there are \\(\\boxed{10}\\) of these.", "6.14 Dividing by \\(2^4 \\cdot 3^2\\) isolates n and makes computation simple:" ], "candidateMarkers": [ { "id": "6.13", "label": null, "offset": 27 }, { "id": "6.14", "label": null, "offset": 206 }, { "id": "6.15", "label": null, "offset": 441 }, { "id": "6.16", "label": null, "offset": 952 }, { "id": "6.17", "label": null, "offset": 1235 }, { "id": "6.18", "label": null, "offset": 1551 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0053.png", "sha256": "6e9441bcdac0b0416e257fa74a27a1c03d4d38278d23609e86272485a3474148", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 54, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2434f96d061fa6865255532cc29df68a7c4acc3b8bb692741d19f4baa6ef8fba", "textHash": "6eff4df95e62a801159155824751d759d5d4cb35c00c2f9d4ff03058a1576cee", "characters": 2218, "flags": [], "candidateHeadings": [ "Review Problems", "6.19 Every multiple of 3 less than or equal to 60 contributes at least one factor of 3 to 60! Every multiple of 3² = 9 that is less than or equal to 60 contributes one more additional factor of 3 to 60! Every multiple of 27 contributes yet one more additional extra factor of 3 to 60!", "6.20 It's easiest to sum the proper divisors from greatest to least. As soon as the sum exceeds the integer itself, we know the number is abundant.", "6.21 In an even three-digit palindrome, the first and the last digit are the same even nonzero digit (the palindrome cannot start with 0). There are 4 such digits. For each of these 4 digits, the middle digit can be any of the 10 digits from 0 to 9 inclusive. The total number of even, three-digit palindromes is 4.10 = \\(\\underline{40}\\).", "6.22 A five-digit palindrome has digits in the form abcba. Since the first digit cannot be 0, there are 9 choices for a. There are 10 choices for each of b and c. Each different choice of a, b, and c creates a distinct five-digit palindrome, so there are a total of 9. 10. 10 = \\(\\underline{900}\\) of them.", "6.23 In Problem 6.5.4, we saw that 66 is the smallest abundant palindrome. In Problem 6.20, we saw that 77 is deficient, so we test 88:" ], "candidateMarkers": [ { "id": "6.19", "label": null, "offset": 84 }, { "id": "6.20", "label": null, "offset": 697 }, { "id": "6.21", "label": null, "offset": 1313 }, { "id": "6.22", "label": null, "offset": 1654 }, { "id": "6.23", "label": null, "offset": 1962 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0054.png", "sha256": "76b5c3136129a2122cc7dbe038753567e3579fe47ad51211efb6770786a843ce", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 55, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "687f44e99d15df24b87e66905cce59b99e69347d72171f46068682fba56697b0", "textHash": "76285d188789c1e9ca3e4521977cfb5c5a461266d5381e1c81255d3519b14701", "characters": 2954, "flags": [], "candidateHeadings": [ "Challenge Problems", "6.24 Because of the symmetric nature of the number 11, it is a divisor of many palindromes. Expanding powers of \\(x + y\\) (where \\(x = 10\\) and \\(y = 1\\)) helps us see why the first few powers of 11 are all palindromes:", "6.25 First, we note that since \\(n\\) has exactly 4 positive divisors that its prime factorization is either in the form \\(a^3\\) or \\(b^3 \\cdot c^3\\), where \\(a\\), \\(b\\), and \\(c\\) are all prime.", "6.26 Since \\(100^{100} = (10^2)^{100} = 10^{200}\\), we know that \\(100^{100}\\) expanded has 201 digits: a 1 followed by 200 0's." ], "candidateMarkers": [ { "id": "6.24", "label": null, "offset": 47 }, { "id": "6.25", "label": null, "offset": 707 }, { "id": "6.26", "label": null, "offset": 1644 }, { "id": "6.27", "label": null, "offset": 2504 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0055.png", "sha256": "a7193ee42b017308002cb5f0f60bcb4824d31345341725619a2e60bec40ccf16", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 56, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "c62caa56804da4fa527ea6a85666f470c513cba2bd5eda97ce802c3c7e3dce20", "textHash": "c626fc3d5db4a1031d2c8609a52dd6ac383847d9bb1ab3c71acc3337c1587cd6", "characters": 2468, "flags": [], "candidateHeadings": [ "Challenge Problems", "6.28 The only proper divisor of a prime number \\(p\\) is 1, so \\(s(p) = 1 < p\\), meaning every prime is deficient.", "6.29 Since \\((n + 1)! = (n + 1)n!\\) for any positive integer \\(n\\), we can relate nearby factorials without a lot of computation:", "6.30 Five-digit palindromes are in the form \\(abba\\), while six-digit palindromes are in the form \\(abcca\\). In either case, there are 9 possibilities for \\(a\\) (1 through 9), and 10 possibilities for each \\(b\\) and \\(c\\), for a total of \\(9 \\cdot 10 \\cdot 10 = 900\\) five-digit palindromes and 900 six-digit palindromes. Neither total is greater.", "6.31 First, note the formula for the sum of the first \\(n\\) positive integers:" ], "candidateMarkers": [ { "id": "6.28", "label": null, "offset": 214 }, { "id": "6.29", "label": null, "offset": 329 }, { "id": "6.30", "label": null, "offset": 640 }, { "id": "6.31", "label": null, "offset": 989 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0056.png", "sha256": "157d943e076d0a6cfb027650e3c9b7163e25a4a91ab1acf6d8fbe6b23561dbfb", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 57, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "3059d2e4d984d43e065096f3995d8ad62156005d84d3d26edde4f7c6b6ac96c3", "textHash": "23a534e5fa175d3813805a9eb9f04aa87fa3213de573fa6814e06864d2c1c2eb", "characters": 2656, "flags": [], "candidateHeadings": [ "6.32 The only prime that is a multiple of 3 is 3. This means that any pair of twin primes other than 3 and 5 must be two integers that are 1 more and 1 less than a multiple of 3. If the smaller one is 1 more than a multiple of 3, then the larger one is a multiple of 3, which is impossible. This means that a pair of twin primes greater than 3 and 5 must be in the form \\(3n-1\\) and \\(3n+1\\) for some positive integer n. Their sum is \\(6n\\), and \\(3|6n\\), so the sum of any pair of twin primes other than 3 and 5 is a multiple of 3.", "6.33 As we look for clues that relate the sum of two three-digit palindromes to a four-digit palindrome, we note that since the three-digit palindromes are both less than 1000, their sum is less than 2000. This means the four-digit palindrome is in the form \\(1aa1\\) for some digit a. Now, we have a better view of the addition of two three-digit palindromes, bcb and ded:" ], "candidateMarkers": [ { "id": "6.32", "label": null, "offset": 463 }, { "id": "6.33", "label": null, "offset": 997 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0057.png", "sha256": "6ae47b2da67c5b53d52587c598c9070eb6f9abdcce630e16c89bf39383df8b63", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 58, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9b6a14ce768af22d580fd9a50b6d9c45ebff2e70be50ec025d9295c1603135a4", "textHash": "32c3684d1f75f52567d7d94e9d7679828490a5a0a67232ef741ad319afc47e36", "characters": 2530, "flags": [], "candidateHeadings": [ "Challenge Problems", "6.34 A ten-digit palindrome has digits in the form \\(abcddecdba\\). There are 9 possibilities for \\(a\\) (1 through 9), since the ten-digit number cannot start with zero, and 10 possibilities each for \\(b, c, d\\), and \\(e\\) (0 through 9). So, the total number of ten-digit palindromes is \\(9 \\cdot 10^4 = 90000\\).", "6.35 Since \\(8^n = (2^3)^n = 2^{3n}\\), we can solve the problem by first finding the largest power of 2 that divides 100!:", "6.36 \\(T(n)\\) is the number of terminal zeros of \\(n!\\), so \\(10^{T(n)}\\) | \\(n!\\). This means \\(n!\\) has at least \\(T(n)\\) factors of 2 and \\(T(n)\\) factors of 5. Since \\(n!\\) always has at least as many factors of 2 than 5, \\(n!\\) has exactly \\(T(n)\\) factors of 5.", "6.37 We can rewrite the given product as the quotient of two factorials:" ], "candidateMarkers": [ { "id": "6.34", "label": null, "offset": 206 }, { "id": "6.35", "label": null, "offset": 519 }, { "id": "6.36", "label": null, "offset": 1041 }, { "id": "6.37", "label": null, "offset": 2378 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0058.png", "sha256": "d6d443fb7443fd9cd2cce3dd148a40e339ef7b3237674b796121188f0bf6430a", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 59, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "af43fe63b0718f1aa3483fd2efbfcf7aa96b12ce200c42115bd72f082dfd73fc", "textHash": "ebe9d32cb67897c440f2ea552cfca0812dcb2c197a39d934e56ab48b5d56248d", "characters": 1797, "flags": [], "candidateHeadings": [ "6.38 Let \\(p^n\\) be a power of a prime. The proper divisors of \\(p^n\\) are 1, \\(p\\), \\(p^2\\), ..., \\(p^{n-1}\\), so" ], "candidateMarkers": [ { "id": "6.38", "label": null, "offset": 606 }, { "id": "6.39", "label": null, "offset": 907 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0059.png", "sha256": "d6fd6685459fafc18fa1da987b67be2ae3bf0ccec393f6d64271245da1cd53c7", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 60, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f867dd7576ebc9baf115c7e32377e5f08d8776025c215a26397979318af24e43", "textHash": "06dee94b9fad025a282b23ec954c531677d777a7a9b5ecf5b020372d76d7d990", "characters": 3105, "flags": [], "candidateHeadings": [ "Challenge Problems", "6.41 If \\(n\\) is perfect or abundant, then \\(s(n) \\geq n\\). Let \\(d_1, d_2, d_3, \\ldots, d_k\\) be the proper divisors of \\(n\\), so", "6.42 Since \\(12^n = 2^{2n} \\cdot 3^n\\), we are looking for the largest value of \\(n\\) such that \\(2^{2n}\\) and \\(3^n\\) are divisors of 20!", "6.43 It's easier to work with integers than fractions, so we multiply through by an integer that leaves us with only integers. In this case, multiplying by 7! does the trick:" ], "candidateMarkers": [ { "id": "6.40", "label": null, "offset": 19 }, { "id": "6.41", "label": null, "offset": 838 }, { "id": "6.42", "label": null, "offset": 1819 }, { "id": "6.43", "label": null, "offset": 2461 } ], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0060.png", "sha256": "ec7269616dde6d2b36dffdc30761867158813d88f4654af0843f9651bb559f32", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 61, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ca34ec410128fa397b260b904d3c3a76e0ba29c583726915eb76bac967d08fe6", "textHash": "1519ed298768e01df6fbc8e107cd2255ef53330adb1871a03d391a5b3bd377c2", "characters": 1504, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter6", "evidence": "book05-source/answer-0061.png", "sha256": "befad5c980952c33de4263fc09a9e723dbbf3abcd3dd88d965e77d4155404298", "nextChapterHeadingPDF": 140 } }, { "source": "answers-05", "pdfPage": 62, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f2aadfbb092f2879a9dae4590b915afd7a16d73a699c0ebe544f8b2778eb73fa", "textHash": "639103b2586de3cf7b8d5fb84309d345fa1e4d5f66a39d5e9fe7a21a54d1e0ef", "characters": 2037, "flags": [], "candidateHeadings": [ "Challenge Problems", "7.10 Perfect squares greater than 0 have only even exponents in their prime factorizations. Integers that are divisible by 8 include 2 to at least the third power in their prime factorizations. But, since the exponents must all be even, squares that are divisible by \\(2^3 = 8\\) must also be divisible by \\(2^4 = 16\\).", "7.11 Let the number be AB where A is the tens digit and B is the units digit. We are given that", "7.12 We are trying to determine when \\(n^2 - 3n + 2\\) is prime. Since primality is based on the existence of factors between 1 and a quantity, it seems natural to look for an algebraic factorization: \\(n^2 - 3n + 2 = (n - 1)(n - 2)\\). A prime number has only one divisor greater than 1, so the smaller of the two factors n - 1 and n - 2 must be 1. Thus, \\(n - 2 = 1\\) and n = 3 is the only solution and the answer is \\(\\boxed{1}\\)." ], "candidateMarkers": [ { "id": "7.10", "label": null, "offset": 52 }, { "id": "7.11", "label": null, "offset": 1258 }, { "id": "7.12", "label": null, "offset": 1601 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0062.png", "sha256": "54be65d0e4b5b30f4552dfcd37f09c66d00ede3d5faa50507f69ea8210687db9", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 63, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f35c8b22ace4bb8b5c23032b6ea0dbfa7730b9c1d78646ed9860a1ff5a97116f", "textHash": "f119b2518976314fbaa9833814ffed7136e7817fce6cdc67671e740d86b301fd", "characters": 2494, "flags": [], "candidateHeadings": [ "7.13 Let the three two-digit integers we are looking for be \\(ab\\), \\(cd\\), and \\(ef\\) (where \\(a\\), \\(b\\), \\(c\\), \\(d\\), \\(e\\), and \\(f\\) are the values of the digits). Since \\(3^2 + 4^2 + 5^2 = 50\\), we have", "7.14 An integer that is a multiple of 15, 20, and 25 must be a multiple of their LCM, which is 300. We can express such integers in the form \\(300n\\) for integer values of \\(n\\). Our goal is to count the integers in this form such that", "7.15 Let \\(q\\), \\(d\\), and \\(n\\) represent the numbers of quarters, dimes, and nickels in John’s jar, respectively. We express \\(d\\) and \\(n\\) in terms of \\(q\\) to reduce the number of variables we have to work with:", "7.16 Let’s take a look at what happens when we subtract the sum of the digits of a given two-digit number \\(AB\\) (where \\(A\\) and \\(B\\) are digits) from the number itself:" ], "candidateMarkers": [ { "id": "7.13", "label": null, "offset": 33 }, { "id": "7.14", "label": null, "offset": 594 }, { "id": "7.15", "label": null, "offset": 1169 }, { "id": "7.16", "label": null, "offset": 2287 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0063.png", "sha256": "77ab9c55dba99a8d89efa431ad7f0e6c0dcb62f8445a54d10bfa285721783886", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 64, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "b7d2b02b299f197c7f6b64d430dfe072bf45946563156a7baab0f002df03bfd2", "textHash": "d88f02aff7bd28f4d44ae1f39ff058762e0b39e694928fe9d6fc2ca68bdf6000", "characters": 1905, "flags": [], "candidateHeadings": [ "Challenge Problems", "7.17 We are looking for values of n that are at least 3 (because a polygon must have at least 3 sides), such that", "7.18 We know that for a and b and some positive integers m and n that", "7.19 It's easier to work without fractions, so we multiply both sides of the equation by 6xy to get" ], "candidateMarkers": [ { "id": "7.17", "label": null, "offset": 472 }, { "id": "7.18", "label": null, "offset": 1012 }, { "id": "7.19", "label": null, "offset": 1611 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0064.png", "sha256": "3ce1a5b9d6543a661f6a14d37efb906be20ddb7fa734f4fa64b1d157d314e967", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 65, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "08163c756b3d8fea989c3a1902c7b709f88a8ca994dc4526e814af551ed4c7a8", "textHash": "dd4d8591150ed25bcf15b3396e4154a0bf4f717a00f0e82498d7fd6e09a99369", "characters": 2380, "flags": [], "candidateHeadings": [ "7.20 For any decimal digit \\(d\\), we have", "7.21 It is natural for students who don't see the answer right away to explore a bit, so before we get into the solution, let's take a look at some examples:", "3.5 = 15 = 42 - 1", "5.7 = 35 = 62 - 1", "11.13 = 143 = 122 - 1", "17.19 = 323 = 182 - 1", "29.31 = 899 = 302 - 1", "7.22 We apply the Division Theorem to this problem. We are told that when \\(x\\) is divided by \\(y\\) that the quotient is \\(u\\) and the remainder is \\(v\\):", "7.23 A perfect square that is a multiple of both 3 and 4 must have exponents of at least 2 for prime factors 2 and 3, so it must be a multiple of \\(2^2 \\cdot 3^2 = 36\\). These integers can be written in the form \\(36n^2\\) for integers \\(n\\). Since we are looking for three-digit integers, we have the inequality" ], "candidateMarkers": [ { "id": "7.20", "label": null, "offset": 802 }, { "id": "7.21", "label": null, "offset": 1038 }, { "id": "3.5", "label": null, "offset": 1197 }, { "id": "5.7", "label": null, "offset": 1216 }, { "id": "11.13", "label": null, "offset": 1235 }, { "id": "17.19", "label": null, "offset": 1258 }, { "id": "29.31", "label": null, "offset": 1281 }, { "id": "7.22", "label": null, "offset": 1625 }, { "id": "7.23", "label": null, "offset": 2036 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0065.png", "sha256": "36d8aa2cf8d2b6aab67a9552475dab99388b849bd771115907944885e7beb6dc", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 66, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "c04d9823e96b5fcfee4cde001969e3c3edd126abf336419a9a90ef6c999cecd0", "textHash": "5236472b38b7023fd064bea5127bbf3592191d27846106857cbf3d1cdd662883", "characters": 1951, "flags": [], "candidateHeadings": [ "Challenge Problems", "7.24 The first thing we do is change the word problem into a math problem. Let \\(A\\) be the number of pigs the first farmer gives to the second and let \\(B\\) denote the number of goats the first farmer gives to the second. Let \\(n\\) be the total amount of debt settled in the transaction. Note that either \\(A\\) or \\(B\\) can be a negative integer. This would represent the first farmer receiving animals. Our equation is" ], "candidateMarkers": [ { "id": "7.24", "label": null, "offset": 389 }, { "id": "7.25", "label": null, "offset": 1310 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0066.png", "sha256": "1d691f4c788c388d9a6b645e54c8402482157ddab5bee0e0681e9e85541a93f7", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 67, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "64a02322ccb791dcb82f80120f2449d440a7d2cf374951fac2293fe675056043", "textHash": "83779bb66c272371511e4ae32ce7b02a9954833f89db954c84d383cb2314d115", "characters": 2240, "flags": [], "candidateHeadings": [ "7.26 We are given several pieces of information to work with and we’ll have to combine them. We are looking for even integers, so they must have positive powers of 2 in their prime factorizations. The numbers are also squares and cubes, so the exponents in their prime factorizations must be multiples of 2 and 3, and therefore 6. We are looking for integers in the form \\(2^6 n^6 = 64n^6\\) that have four digits:", "7.27 This one turns out to be less difficult than it initially looks. Since we are dealing with products of integers, we take a second look at the equation where the right-hand side is prime factored:", "7.28 We are given that \\(a^2 - b^2 = p\\) for some positive integers \\(a\\) and \\(b\\) and some prime number \\(p\\). We can factor the difference of squares:" ], "candidateMarkers": [ { "id": "7.26", "label": null, "offset": 598 }, { "id": "7.27", "label": null, "offset": 1423 }, { "id": "7.28", "label": null, "offset": 1886 } ], "visualReview": { "status": "scan-compared-chapter7", "evidence": "book05-source/answer-0067.png", "sha256": "9c5ecd799f48c60f95e0174470d1fe656b4702af3acd43f519bfe63319164674", "nextChapterHeadingPDF": 156 } }, { "source": "answers-05", "pdfPage": 68, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "c9745bbe47651a44a1f2160bfbdbfaf7828ce5ec25061425e54f454c0bfb4062", "textHash": "6480e9149b2f1aa325f18b663898054ec314a16feacd1e041e5ff9198cbf9c71", "characters": 446, "flags": [], "candidateHeadings": [ "Exercises for Section 8.2", "Exercises for Section 8.3" ], "candidateMarkers": [ { "id": "8.2.1", "label": null, "offset": 56 }, { "id": "8.3.1", "label": null, "offset": 122 }, { "id": "8.3.2", "label": null, "offset": 167 }, { "id": "8.3.3", "label": null, "offset": 218 }, { "id": "8.3.4", "label": null, "offset": 286 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0068.png", "sha256": "b2648575327e93d09495b4f6b714832edb476af67a992874229223d8244b1aa1", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 69, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "cff011be49bbf3bedbc9e6d02a103888290a1536230e6bca53d79759f5689149", "textHash": "12b810e63a792f1979f4abe3a629d64268d7ab672cf33ee548a35540522549b5", "characters": 1134, "flags": [], "candidateHeadings": [ "Exercises for Section 8.4", "Exercises for Section 8.5" ], "candidateMarkers": [ { "id": "8.3.5", "label": null, "offset": 31 }, { "id": "8.3.6", "label": null, "offset": 242 }, { "id": "8.4.1", "label": null, "offset": 357 }, { "id": "8.4.2", "label": null, "offset": 481 }, { "id": "8.5.1", "label": null, "offset": 710 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0069.png", "sha256": "14bf5028088945b7f902a98b9c4234706db4bca4ced9fbda1d0cb6e8843597ca", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 70, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "d9f90e229ffc664ec1b274c3badca070d9a86e0c39fead25fa607795ba297c10", "textHash": "5f852e4e3c6d015b1fadd24630cfd7ee5eceeafc447b9806075c024ad9df5d43", "characters": 2408, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.5.2", "label": null, "offset": 12 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0070.png", "sha256": "30f5fbe904a89773315bbdeb99f5e77c24853d0a69a67689201a02acef894554", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 71, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "4bdd70df7a5299a268e89f744775b3ac5ca416deb5707863b46804c2e7039952", "textHash": "51da369d121ebb09eda2b75b5d16e413b3ca5817857e19482bf1561a975dd722", "characters": 1601, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.5.3", "label": null, "offset": 31 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0071.png", "sha256": "354c6ca17ab92aaec8158793fb41490d178a31c8e9325f9f138ba1fe732ee598", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 72, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "8ffd50067fa148430766ac8924c0ac86212230c44c2b3f70972e0d07bbfd50fb", "textHash": "ca38dc31aafe8c30d63a03ab15d26cad9614936176d04d973bda362a3696ebed", "characters": 1485, "flags": [], "candidateHeadings": [ "Exercises for Section 8.6" ], "candidateMarkers": [ { "id": "8.5.4", "label": null, "offset": 155 }, { "id": "8.6.1", "label": null, "offset": 368 }, { "id": "8.6.2", "label": null, "offset": 919 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0072.png", "sha256": "9cf47d36f97117674a9111970d8494df896b5574882337c823897bae05945ee4", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 73, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "50774a4b8407a81b05ca3e9d256b4c4814d93bbbe9047e8a68505970b40dd449", "textHash": "1701bd86286f92e24b775dede418ad8db13999715a52c08d6df8a52ee5e707d9", "characters": 948, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0073.png", "sha256": "3a0ddb07c753e04f17db3f63731e16515a7f2cb0814f39357e022312caeed6b7", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 74, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "4d74fe42966ce07566abfc377e582ef9da6c82e2ca23efbc76679c4bb6f0e958", "textHash": "771dfeee273294570e722730d46f9fa44be143f4b789e51c8183991189527607", "characters": 2088, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "8.6.3", "label": null, "offset": 12 }, { "id": "8.6.4", "label": null, "offset": 924 }, { "id": "8.6.5", "label": null, "offset": 1479 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0074.png", "sha256": "5d1c44d55a1e745e2302ffcffc50a97285c808445ad7bae30324d8cb9c8a6729", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 75, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "90fc7cd003a1467b9dbad3e7cf0602472c3c29399f79c40236fbdeb4c857c16d", "textHash": "18cbdf17e60f641785008299b5d970d739bf63a9800ac373df1eb89c111d2629", "characters": 1140, "flags": [], "candidateHeadings": [ "Review Problems" ], "candidateMarkers": [ { "id": "8.17", "label": null, "offset": 47 }, { "id": "8.18", "label": null, "offset": 86 }, { "id": "8.19", "label": null, "offset": 173 }, { "id": "8.20", "label": null, "offset": 664 }, { "id": "8.21", "label": null, "offset": 938 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0075.png", "sha256": "b0fa824b7f44de586af6e9f77892ccf7f4e84a40159d9d675a7f777ff45de7a9", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 76, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "7f3548d7653df5ef1faa032fa9aed3a779f6b1f359565738984c75b1db6731b4", "textHash": "0843c6396ab49d9167067ea1e2e9a8168d42d834c8dd3301751405734c39e8f0", "characters": 905, "flags": [], "candidateHeadings": [ "Review Problems" ], "candidateMarkers": [ { "id": "8.22", "label": null, "offset": 16 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0076.png", "sha256": "8943210d9b5f97e363de59d0024640187df8f58ac5536161ed7f52671027aaf0", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 77, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "454d43c7ed36e1b4139a5bc2c950bf88ce0237640f55a6d8ed238fd9c8b6d9fa", "textHash": "0c6fe494b37e5c54b61f059a30501592217a6bf0408878750e7d348f8d77ce95", "characters": 11722, "flags": [ "truncated" ], "candidateHeadings": [ "8.23 We are looking for the smallest base \\(b\\) such that \\(100_b \\leq 62 < 1000_b\\), which is the same as saying that \\(b^2 \\leq 62 < b^3\\). The smallest perfect cube greater than 62 is 64, so the smallest possible value of \\(b\\) is \\(\\sqrt[3]{64} = \\boxed{4}\\).", "Challenge Problems", "8.24 The number of terminal zeros in which an integer ends in base 12 is the number of factors of 12 that divide that integer. Since \\(12 = 2^2 \\cdot 3^1\\), we must find the number of factors of both 2 and 3 in 10! in order to find the number of factors of 12.", "8.25 The goal is to count in base 3 using only binary digits. The \\(100^th\\) smallest positive binary integer is \\(100 = 1100100_2\\), so the \\(100^th\\) smallest positive integer that can be written with only the binary digits is \\(1100100_3 = \\boxed{981}\\).", "8.26 It's easier to count the integers that don't have a 7 digit than it is to count those that do, so we use complementary counting to solve this problem.", "8.27 The smallest positive integer that needs 4 digits when expressed in base \\(b\\) is \\(1000_b = b^3\\). Therefore, the largest integer that can be expressed in base \\(b\\) using 3 digits is \\(b^3 - 1\\).", "8.28 Since \\(15 = 3^1 \\cdot 5^1\\), the largest possible value of \\(n\\) for which \\(15^n = 942!\\) is the largest possible value of \\(n\\) for which both \\(3^n = 942!\\) and \\(5^n = 942!\\). Since \\(942!\\) has many more factors of 3 than it does 5, our answer will be the number of factors of 5 in \\(942!\\)." ], "candidateMarkers": [ { "id": "8.23", "label": null, "offset": 31 }, { "id": "8.24", "label": null, "offset": 316 }, { "id": "8.25", "label": null, "offset": 996 }, { "id": "8.26", "label": null, "offset": 1255 }, { "id": "8.27", "label": null, "offset": 1838 }, { "id": "8.28", "label": null, "offset": 2163 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0077.png", "sha256": "ac232e50edac2fc7f2a8f4272fead167c23c1b421bba88d533bbdd2de3eabbbe", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 78, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "318439cc3c90805de6980f597af550b9fcae46aa14485220cd868093a7a5f9db", "textHash": "4e6ad7019d8b188162c9b2901509f0d988dc403e6fdd1103e58a13f16ed66688", "characters": 3068, "flags": [], "candidateHeadings": [ "Challenge Problems", "8.29 The answer is \\(\\underline{\\text{no}}\\).", "8.30 Since \\(3^{2} = 9\\), we can convert directly from base 3 to base 9 by converting each pair of digits (starting with the rightmost digits) to base 9. Since \\(x = 121122111222111122223\\), the leftmost digit in base 9 is the conversion of the leftmost digit pair from base 3, to base 9. Since \\(12_{3} = 5_{9}\\), the answer is \\(\\underline{\\text{5}}\\).", "8.31 Since \\(23_{b} = 2b + 3\\) and \\(b > 3\\), \\(23_{b}\\) can be any odd integer greater than 2(3) + 3 = 9. We are looking for the next smallest odd perfect square, which is \\(5^{2} = 25\\). Since \\(2b + 3 = 25\\), \\(b = \\underline{\\text{11}}\\) is our answer.", "8.32 An eight-bit binary word is a nonnegative integer with less than 9 digits when expressed in binary. This means we are counting the integers from 0 to \\(2^{8} - 1\\) inclusive, of which there are \\(2^{8} = \\underline{\\text{256}}\\). We could also view each digit as a \"choice\". There are 2 choices for each of 8 digits, so \\(2^{8}\\) eight-bit binary words are possible.", "8.33 We can count 1 down from 32768 = 100000\\(_{8}\\) to get \\(\\underline{\\text{777778}}\\).", "8.34 We are looking for integers n such that", "8.35 It might at first be tempting to think that we can create a correspondence between the three digits we are using and the three digits used in base 3. However, this correspondence is made difficult by the fact that the smallest base-3 digit 0 cannot be the lead digit of a positive integer, whereas all three of 1, 3, and 5 can." ], "candidateMarkers": [ { "id": "8.29", "label": null, "offset": 19 }, { "id": "8.30", "label": null, "offset": 307 }, { "id": "8.31", "label": null, "offset": 662 }, { "id": "8.32", "label": null, "offset": 919 }, { "id": "8.33", "label": null, "offset": 1291 }, { "id": "8.34", "label": null, "offset": 1382 }, { "id": "8.35", "label": null, "offset": 1661 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0078.png", "sha256": "aad7703c2daef270f4d1c81c0466e2817545df53aecb7b4d0d1fcad8b6c5c4cd", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 79, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "fad6487ccdab2df24dcf197bebb7e375bce2e0f79754a6848fd303b97df1b8b6", "textHash": "ab8522e6f3b6852d0ef32500568bebf51d074b3806e2d7ad9027017272b338de", "characters": 2452, "flags": [], "candidateHeadings": [ "8.36 Since \\(47_{a} = 74_{b}\\), we know that \\(4a + 7 = 7b + 4\\) where a and b are each at least 8 (since the largest digit of each base number is 7). Since \\(b \\geq 8\\), the smallest possible value of the integer represented as \\(47_{a}\\) and \\(74_{b}\\) is \\(7 \\cdot 8 + 4 = 60\\). However, \\(4a + 7 = 4(a + 1) + 3\\), so we are looking for an integer that leaves a remainder of 3 when divided by 4. We try b = 9 and get 7b + 4 = 67. Since \\(4a + 7 = 67\\) has a solution a = 15, we have a pair of bases that work.", "8.37 \\(343 = 7^{3} = 1000_{7}\\), so the first 343 positive integers in base 7 are \\(17, 27, \\ldots 1000_{7}\\). Any number in this list that neither includes 4 or 5 only includes the digits 0, 1, 2, 3, and 6. If we replace 6 with 4, these have the same decimal expansions as the integers in base 5. Since there are \\(5^{3} = 125\\) positive integers less than or equal to 1000, there are 125 integers less than or equal to 1000; that contain no 4's or 5's in base 7, which means there are \\(343 - 125 = 218\\) integers that include a 4 or a 5." ], "candidateMarkers": [ { "id": "8.36", "label": null, "offset": 31 }, { "id": "8.37", "label": null, "offset": 831 }, { "id": "8.38", "label": null, "offset": 1373 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0079.png", "sha256": "1b184c8203d87e86f9e5a46b2cef2bb0bc16080aec3335f930d0bfac28a95582", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 80, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "cafb5002f6bfb3bc306d577fd73bfd970952508de0237b6a6f482c85aa6cffa8", "textHash": "dd9e731a34b34f72e1a00e4d004e94591b671a1e513cf616ddd73e69c110af61", "characters": 3246, "flags": [], "candidateHeadings": [ "Challenge Problems", "8.39 Let a, b, and c be digits such that \\(N = abc_7 = cba_9\\). This gives us the equation", "8.40 We can use binomial expansion to more easily convert each power of \\(10b_{+1}\\) to each given base b:", "8.41 The integers that can only be written using 0, 1, 2, 3, 5, 6, 7, 8, 9 in base 10 correspond with the integers that are written with 0, 1, 2, 3, 4, 5, 6, 7, 8 in base 9, where 5 in base 10 corresponds with 4 in base 9, 6 in base 10 corresponds with 5 in base 9, etc. So 2005 on the odometer is the same as 2004, which is \\(\\boxed{1462}\\) miles in base 10.", "8.42 Since the integer has 2 terminal zeros in base 3, it is a multiple of \\(3^2\\). Since it has a terminal zero in each base 4 and base 5, it is a multiple of each 4 and 5. This means the integer is a multiple is \\(lcm[9, 4, 5] = 2^2 \\cdot 3^2 \\cdot 5^1 = 180\\). If the integer has a terminal zero in base b, then it is a multiple of b. Since the integer must have \\(t(180) = 18\\) positive divisors (possibly others), there are \\(18 - 4 = \\boxed{14}\\) other bases (other than 1, 3, 4, and 5) in which the integer must have at least one terminal zero.", "8.43 Since \\(VYZ\\), \\(VYX\\), and \\(VWW\\) are three consecutive base-5 integers, their units digits must be 3, 4, and 0 respectively because their middle digit changed between the second and third of these numbers. This means that \\(Z = 3\\), \\(X = 4\\), and \\(W = 0\\) and the three consecutive integers are \\(VY3\\), \\(VY4\\), and \\(VVO\\). Additionally, \\(V = Y + 1\\), because when the middle digit went up, it couldn't have gone up from 4 to 0 because those digits are already used by other letters. This means \\(V = 2\\) and \\(Y = 1\\), so \\(XYZ_5 = 413_5 = \\boxed{108}\\).", "8.44 Base-3 integers that don't use the digit 2 correspond to binary integers. We need to count all the integers up to 1992 that have a base-3 representation that looks like binary, so we start by converting 1992 to base 3: \\(1992 = 2201210_3\\). The largest integer less than this that has only digits 0 and 1 is \\(1111111_3\\). This corresponds to the binary integer that is the last we need to count: \\(1111111_2 = 2^7 - 1 = \\boxed{127}\\)." ], "candidateMarkers": [ { "id": "8.39", "label": null, "offset": 19 }, { "id": "8.40", "label": null, "offset": 443 }, { "id": "8.41", "label": null, "offset": 1317 }, { "id": "8.42", "label": null, "offset": 1678 }, { "id": "8.43", "label": null, "offset": 2231 }, { "id": "8.44", "label": null, "offset": 2801 } ], "visualReview": { "status": "scan-compared-chapter8", "evidence": "book05-source/answer-0080.png", "sha256": "6b85d71d975f7b6394c11a05ec62acdc5ba0353681bf81050336d7d897f0852b", "nextChapterHeadingPDF": 180 } }, { "source": "answers-05", "pdfPage": 81, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "dc02cf35999080f458739d174be974ba2556aa193aa4a77fe56a540feab4cdeb", "textHash": "3d80dce6fb9ddbf10ddee5f0134d741c10dd55dc6b0b3179f61e39d460aa2dd1", "characters": 378, "flags": [], "candidateHeadings": [ "Exercises for Section 9.2" ], "candidateMarkers": [ { "id": "9.2.1", "label": null, "offset": 93 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0081.png", "sha256": "72384135043a08f5ccdc150c480a0eb69738d3837fc48f891a60daf2d2aa5a07", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 82, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ed3a41928b973e2b13e3fab2a00393329eb739318bb46a52108a5eb13c5289fe", "textHash": "56178000f050f15a3522f0f00c48dd89221d2242777c3a6e6a85cd725536287c", "characters": 1098, "flags": [], "candidateHeadings": [ "Exercises for Section 9.3" ], "candidateMarkers": [ { "id": "9.3.1", "label": null, "offset": 39 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0082.png", "sha256": "4bd057319925e14030d3ce8fdfd850df43576cd66f85269eb25b144cafaecb7e", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 83, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "81658b7573e6d594899cbb1c3078b78a07e86b5f25a5917326efbe06298974c2", "textHash": "69c240f9e677f36aff1e096483dc0337d072007de55e4a3ca1ba821ed5b83653", "characters": 1180, "flags": [], "candidateHeadings": [ "Exercises for Section 9.4", "Exercises for Section 9.5" ], "candidateMarkers": [ { "id": "9.4.1", "label": null, "offset": 60 }, { "id": "9.5.1", "label": null, "offset": 291 }, { "id": "9.5.2", "label": null, "offset": 1048 }, { "id": "9.5.3", "label": null, "offset": 1113 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0083.png", "sha256": "d1f0509bfe36a32798ec4b0ed9ae7ca5e52d0284aa031d20874011bfd0f122bc", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 84, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "3392a826ee439ff5ba095a67c97aa0cca7848038c1fd3f428d96ff432d3884d6", "textHash": "433b10c1dc7ba873d2c1e93c352ad4cff7d4085ac0dcb694ce11dd046f8669c1", "characters": 8135, "flags": [ "repeated-ocr-lines", "truncated" ], "candidateHeadings": [ "Review Problems", "Review Problems", "9.11 (a) \\(42_9\\)" ], "candidateMarkers": [ { "id": "9.5.4", "label": null, "offset": 16 }, { "id": "9.11", "label": null, "offset": 1476 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0084.png", "sha256": "f820498390bf8bb42367b6d58fc67a1b555737f5bfaac82c4a9e8761e148f322", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 85, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "8d9d75b839e27b0a93db1846f7477d8864bcd10cbb2189fd7e358d18e6ac23bd", "textHash": "dfa4214cb5d1fd2ca282e87d1e1faca15f96f3c5d848ce6d09eb752795a26cbf", "characters": 2563, "flags": [], "candidateHeadings": [ "9.16 All of these problems can be solved by converting the integers to base 10 and using regular division or even using base number division. However, we look for methods that can be applied more easily once they are fully understood. Make sure you see why each method works!" ], "candidateMarkers": [ { "id": "9.15", "label": null, "offset": 34 }, { "id": "9.16", "label": null, "offset": 902 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0085.png", "sha256": "8c2ca5b1b8ac97a4f198a07bf5fc51b5300374cb17f18ebee4ec2d2bbd1a4baa", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 86, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "bf8ea9c47de52277ddfa60bc9be8bd8ece21926343d766889e8815da2a7f308b", "textHash": "9cab4824b24de01f4df662687c51cc6237af8e163fd268a840a6a0323057bf67", "characters": 7111, "flags": [ "truncated" ], "candidateHeadings": [ "Challenge Problems", "Challenge Problems", "9.17 In the two rightmost columns of addition, there is no carrying, but in the third, there is, so \\(6b + 1b = 10b\\) and \\(b = 7\\).", "9.18 This problem seems at first like a bigger headache than it really is. However, starting patiently by adding the binary numbers in the parentheses quickly makes the job easier:", "9.19 When we rewrite the base numbers as sums of digit bundles, we get the equation", "9.20 We know \\(b > 4\\) since the digit 4 appears. For every \\(b > 4\\), it's true that \\(11_b \\cdot 313_b = 3443_b\\). There are a number of ways in which we can see why this is true. Straightforward multiplication requires no carrying and long division reveals a similar result.", "9.21 Note that", "9.22 There are a number of ways to approach this problem. Here, we focus on one possible solution that highlights algebraic methods and a bit of arithmetic creativity." ], "candidateMarkers": [ { "id": "9.17", "label": null, "offset": 39 }, { "id": "9.18", "label": null, "offset": 173 }, { "id": "9.19", "label": null, "offset": 554 }, { "id": "9.20", "label": null, "offset": 816 }, { "id": "9.21", "label": null, "offset": 1362 }, { "id": "9.22", "label": null, "offset": 1682 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0086.png", "sha256": "dbdceb9dee37fc874fb342f3f846e73cf326f2fd51657da4c4823716812fb43f", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 87, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ecb49a8cf581abb79dc0a768c55362b0917731175b414785c6fcce775225f9da", "textHash": "d5689cea128dd7d68f266756d53511780358403a2c3792116abde5fae55090b6", "characters": 2896, "flags": [], "candidateHeadings": [ "9.23 We are given that \\(12_b \\cdot 15_b \\cdot 16_b = 3146_b\\). We can express each integer in terms of base-b digit bundles, giving us a polynomial equation:", "9.24 Let the digits of the integer from left to right be \\(d_1, d_2, \\ldots, 7, 1, 7, 7, 7\\). We can write the integer as a sum of base-12 digit bundles:", "9.25 Berris must come up with a method to distinguish the three two-digit numbers. Base 100 to the rescue! Since x, y, and z are distinct base-100 digits and the base-100 number \\(xyz_{100}\\) is unique, Berris can let \\(A = 100^2\\), \\(B = 100^1\\), and \\(C = 100^0\\) so that" ], "candidateMarkers": [ { "id": "9.23", "label": null, "offset": 579 }, { "id": "9.24", "label": null, "offset": 1518 }, { "id": "9.25", "label": null, "offset": 2172 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0087.png", "sha256": "c5b0a14f1605cbfbf2e3e766d559bd46c853a070102620e49150e1aa3f8bfc43", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 88, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2f86f9e73ef3939324dade4d2c56af8137cf1f9d2324e5647e9a63b4be14af67", "textHash": "e6689cdcbf51953d174aecef91e345896a0bcbd060c7dd2dbba81c8c4b05fc0a", "characters": 880, "flags": [], "candidateHeadings": [ "Challenge Problems", "9.26 Finding the sum of the three-digit base-5 palindromes directly seems tedious, so we look for a nicer method. The sum is equal to the total number of them multiplied by their average." ], "candidateMarkers": [ { "id": "9.26", "label": null, "offset": 19 } ], "visualReview": { "status": "scan-compared-chapter9", "evidence": "book05-source/answer-0088.png", "sha256": "19664d121ff6225a43249d588fd1ddf0412bde1186e08ccc71d49493f88d8f45", "nextChapterHeadingPDF": 192 } }, { "source": "answers-05", "pdfPage": 89, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "fc530f46459b09bad53f0915a2945a725339eb0b7c715f23fe24e4db1e157fce", "textHash": "2fb77f742e4cbcbb891dad397def0ef0702c51f803e1865b24c4c5691422d92e", "characters": 1394, "flags": [], "candidateHeadings": [ "Exercises for Section 10.2" ], "candidateMarkers": [ { "id": "10.2.1", "label": null, "offset": 76 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0089.png", "sha256": "4134f04beca83ec2b62d8a042d65616cb296e5e67278dc80f652c0631b6bea3a", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 90, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "bed854c288f174556843d6d4e2c00821b8e2e914f5fe99a822be237aecb0de9e", "textHash": "725555b0765dc612b0bdab55ebcf393266c2d6b8313aa1fb890b7c1f471afb29", "characters": 1776, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "10.2.2", "label": null, "offset": 1383 }, { "id": "10.2.3", "label": null, "offset": 1567 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0090.png", "sha256": "29b904aab14ea5da7deb94ed1ec222dd08686827f02f0c7efd259fd49245d6f0", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 91, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "bff37e522686e3f150163b16b42231804a30d4da08e7c1a21a33832aea515c07", "textHash": "35229ec96f71195919dea2c045c9bd0c3d8af17f7cfbf91384fd5a19213baea4", "characters": 1812, "flags": [], "candidateHeadings": [ "3.1 = 3 Let x be the number with units digit 3, and let y be the other positive integer. Since xy ends in a 1, y must end in a 7, so x + y = 10. The units digit of x + y is \\(\\boxed{0}\\). Notice that we only needed to check odd values of y because the product was odd.", "3.9 = 27", "Exercises for Section 10.3" ], "candidateMarkers": [ { "id": "10.2.4", "label": null, "offset": 25 }, { "id": "10.2.5", "label": null, "offset": 288 }, { "id": "3.1", "label": null, "offset": 297 }, { "id": "3.9", "label": null, "offset": 566 }, { "id": "10.2.6", "label": null, "offset": 576 }, { "id": "10.3.1", "label": null, "offset": 1047 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0091.png", "sha256": "13aa6280505a3a9cd4df89169a7c8daa6a4721f4dc74d2768a3aac2be91d15be", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 92, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "261d0333da8526be23a11a88b39dc16a0ad38c8afd330bfb92e1dabbf3e2356b", "textHash": "304fcf6a411076c2ca55fecdf22980eec17d0e6d59ae36e8ba92ac32ea92fed7", "characters": 1960, "flags": [], "candidateHeadings": [ "Exercises for Section 10.4" ], "candidateMarkers": [ { "id": "10.3.2", "label": null, "offset": 318 }, { "id": "10.3.3", "label": null, "offset": 597 }, { "id": "10.4.1", "label": null, "offset": 1055 }, { "id": "10.4.2", "label": null, "offset": 1277 }, { "id": "10.4.3", "label": null, "offset": 1608 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0092.png", "sha256": "8a76641da1b9c5ce42a0a3acd6a9051aee7e84c7c4968899f33ee9b76d40b52b", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 93, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "fabe94a5765196d9b90fc584cc9060d9a2e648ad9078e5fd01c8a905be2b3a56", "textHash": "9cbf9741c26863ba5047ff1f2c5ed206e35ac47e398ae87f28fe57f1e64d129b", "characters": 2158, "flags": [], "candidateHeadings": [ "Review Problems", "10.20 The units digit of powers of 2 and 3 both repeat in cycles four long. Since \\(1986 \\div 4\\) has a remainder of 2, the units digit of \\(3^{1986} - 2^{1986}\\) is the same as \\(3^{2} - 2^{2} = \\boxed{5}\\).", "10.21 Since Jenny's new system of arranging stamps includes 10 on each page, the number of the last page will be the units digit of her total number of stamps (in base 10). That units digit is the same as the units digit of \\(8 \\cdot 2 \\cdot 6 = 96\\), which is \\(\\boxed{6}\\).", "10.22 Since the product is positive, we need only check the possible odd units digits and we find that \\(\\boxed{7}\\) is the only one that works: \\(7 \\cdot 7 = 49\\).", "10.23 First, we note that the product is odd, so neither of the primes is 2. Also, since the units digit of the product is neither 0 nor 5, the product is not a multiple of 5. Since all integers with units digits of 5 are multiples of 5, we don't need to worry about 5 as a possible units digit. We test the remaining cases. We need only test 10 cases since we can organize the products with the smaller units digit first:" ], "candidateMarkers": [ { "id": "10.4.4", "label": null, "offset": 25 }, { "id": "10.19", "label": null, "offset": 655 }, { "id": "10.20", "label": null, "offset": 777 }, { "id": "10.21", "label": null, "offset": 986 }, { "id": "10.22", "label": null, "offset": 1262 }, { "id": "10.23", "label": null, "offset": 1427 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0093.png", "sha256": "acc45d3674c9115d6ad8b032e6f62fe93ff77f31356911312a1287e99f85936e", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 94, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "1c592f63d1f0dbd38542862450a4aba4ca2b5cf87f7dfeea34c2017a705ff0cf", "textHash": "e7e13c50b79485b497719da49e1ef609dd9a76069908d5aa4995d43d62ff92b4", "characters": 1281, "flags": [], "candidateHeadings": [ "Review Problems", "10.28 First, we simplify the given factorial fraction:", "0.1 = 0 5.6 = 30", "1.2 = 2 6.7 = 42", "2.3 = 6 7.8 = 56", "3.4 = 12 8.9 = 72", "4.5 = 20 9.10 = 90" ], "candidateMarkers": [ { "id": "10.24", "label": null, "offset": 16 }, { "id": "10.25", "label": null, "offset": 85 }, { "id": "10.26", "label": null, "offset": 330 }, { "id": "10.27", "label": null, "offset": 464 }, { "id": "10.28", "label": null, "offset": 772 }, { "id": "0.1", "label": null, "offset": 1016 }, { "id": "1.2", "label": null, "offset": 1035 }, { "id": "2.3", "label": null, "offset": 1053 }, { "id": "3.4", "label": null, "offset": 1071 }, { "id": "4.5", "label": null, "offset": 1090 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0094.png", "sha256": "11d9bab4b4a056049e729f9bf740eb6580323bdf83458fa33b2932a7d7bfedb8", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 95, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "44416c730514c668ac371261756edee09ba1877b4deaafde42761d7e959ad887", "textHash": "e93f7c34278559786b1ea84eddd7600a2ab1387a4e80241d94a452daad1baa38", "characters": 3087, "flags": [], "candidateHeadings": [ "Challenge Problems", "10.30 Since \\(n^4 = (n^2)^2\\), we are looking for the possible base-5 units digits of squares of squares. From Problem 10.26, we know that squares have base-5 units digits of 0, 1, or 4. We also know that the squares of such integers have base-5 units digits of 0, 1, or 1 respectively, so \\(\\underline{0}\\) and \\(\\underline{1}\\) are the only possible units digits of perfect fourth powers in base 5. Notice that this means perfect fourth powers leave remainders of only 0 or 1 when divided by 5.", "10.32 The units digit of \\((133^{13})^{3}\\) is the same as the units digit of \\((3^{13})^{3} = 3^{39}\\). As we've seen, the units digits of powers of three repeat in a cycle of 4. Since \\(39 = 4 \\cdot 9 + 3\\), the units digit of \\(3^{39}\\) is the same as that of \\(3^{8} = 27\\), which is \\(\\underline{7}\\).", "10.33 The pages are consecutive integers and in Problem 10.28 we saw the possible units digits of products of consecutive integers. We sum the units digits of the consecutive integers in both cases where the product has a units digit of 6: \\(2 + 3 = 5\\) and \\(7 + 8 = 15\\) have the same units digit, \\(\\underline{5}\\).", "10.34 As we've seen already, the units digits of powers of 2 repeat in a cycle of 4. Since there are powers of 2 with units digit 6, we know that \\(100 \\div 4 = \\underline{25}\\) of the given powers of 2 have units digits of 6.", "10.35 The units digit of \\(mn\\) is \\(16 = 1\\). Searching for a units digit for \\(n\\) (which is clearly odd), we find that \\(7 \\cdot 3 = 1\\), so \\(\\underline{3}\\) is the units digit of \\(n\\).", "10.36 First we note that the units digit of \\(m^2n = 39^{39}\\) is the same as the units digit of \\(9^{39}\\). The units digits of powers of 9 repeat in a cycle of 2, so this is the same as the units digit of \\(9^1 = 9\\). Since the units digit of \\(n\\) is 1, we know the units digit of \\(m^2\\) is 9. Searching through the possibilities, we know that the units digit of \\(m\\) is either \\(\\underline{3}\\) or \\(\\underline{7}\\)." ], "candidateMarkers": [ { "id": "10.29", "label": null, "offset": 194 }, { "id": "10.30", "label": null, "offset": 827 }, { "id": "10.31", "label": null, "offset": 1324 }, { "id": "10.32", "label": null, "offset": 1618 }, { "id": "10.33", "label": null, "offset": 1925 }, { "id": "10.34", "label": null, "offset": 2244 }, { "id": "10.35", "label": null, "offset": 2471 }, { "id": "10.36", "label": null, "offset": 2662 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0095.png", "sha256": "754d1eb1868c55fb462cc66b9462023106841d6fad4db4b0aebdbb357d7da3d0", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 96, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "76b1a7ccb78969f9276831f91d149b05acd383bc171dcbcbdde2521237999dcd", "textHash": "7ace3952e0e3177c20a50c019dfd40503cad77644f5140ef25f6cbce413c8411", "characters": 2576, "flags": [], "candidateHeadings": [ "Challenge Problems", "10.38 An integer that is a multiple of both 4 and 6 is a multiple of \\(lcm[4, 6] = 12\\). Our goal is to count the three-digit multiples of 12 with units digits of 2. For a positive integer \\(n\\), \\(12n\\) has a units digit of 2 when \\(n\\) has a units digit of 1 or 6. In other words, \\(n\\) is 1 more than a multiple of 5. We can rewrite \\(n\\) as \\(5m + 1\\) for nonnegative integers \\(m\\).", "10.39 A group of 10 consecutive positive integers includes an integer that has each units digit exactly once. Their product has the same units digit of the product of their units digits, which is 0. Their sum" ], "candidateMarkers": [ { "id": "10.37", "label": null, "offset": 19 }, { "id": "10.38", "label": null, "offset": 1505 }, { "id": "10.39", "label": null, "offset": 2340 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0096.png", "sha256": "1a3e190827471ff0894cc6cde048f2eda0f5d743d21278391885304546227789", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 97, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "adc28de3f34308fb6f57d6794d7a0c7bf7ec641f14e9bad911701b81165e980c", "textHash": "f59e9ba48b5afb805c946f3cddaad1c04981554961cd88af1d11a9fa7432ae2a", "characters": 2202, "flags": [], "candidateHeadings": [ "10.41 Cubing each units digit, we find that a cube has a units digit of 3 only when the units digit of its cube root has a units digit of 7, so B = 7. Also, we note that", "10.43 Let M be Margaret's age and W be Wylia's age. We know that M - 2 = W, and that W + M = (M - 2) + M = 2M - 2 has a units digit of 6, so 2M has a units digit of 8. The only possible units digits" ], "candidateMarkers": [ { "id": "10.40", "label": null, "offset": 167 }, { "id": "10.41", "label": null, "offset": 1107 }, { "id": "10.42", "label": null, "offset": 1370 }, { "id": "10.43", "label": null, "offset": 1999 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0097.png", "sha256": "07766b7ddfd5edd5d3a64a6d26a3e48858c9847b12e77dfce4334ad0284d5cf5", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 98, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f886c5005265f61237d458b42a665bbc8c8b5810a8ea61a85521615ce2462e91", "textHash": "ba6e25aec5f9db07e2fc28f0dc6d8dfa1137cb72d4cb097dca8cdef9e9c2fc17", "characters": 2384, "flags": [], "candidateHeadings": [ "Challenge Problems", "10.44 Let N be the integer we seek. We know that N is 2 more than a multiple of each of 3, 4, 5, and 6, so N - 2 is a multiple of \\(\\lfloor \\text{cm} \\rfloor 3\\), 4, 5, 6] = 60. Since N > 2, the number we seek is N = 62.", "10.45 The very first thing we notice is that b ≥ 4 because we are using digits up to 3 already. Our sum is really a product of 7 and 32b, so we know that 7 · 2 = 14 has a units digit of 2 in base b. This means 14 = nb + 2 for some integer n and nb = 12 means that b | 12. The divisors of 12 greater than 3 are 4, 6, and 12, and all 3 satisfy the problem statement.", "10.46 Mersenne primes are \\(\\frac{2}{3} \\times 2^p - 1\\) for prime values of p. If a particular Mersenne prime has a unit’s digit of 3, then 2^p has a unit’s digit of 4. However, in the repeating cycle of units digits of powers of 2, only the powers of 2 in the form \\(2^{4n+2}\\) for some nonnegative integer n have units digits of 4. Since \\(4n + 2 = 2(n + 1) = p\\), 2 | p, so p = 2. This proves there is only 1 Mersenne prime with a unit’s digit of 3, namely, \\(2^2 - 1 = 3\\)." ], "candidateMarkers": [ { "id": "10.44", "label": null, "offset": 130 }, { "id": "10.45", "label": null, "offset": 352 }, { "id": "10.46", "label": null, "offset": 718 }, { "id": "10.47", "label": null, "offset": 1198 } ], "visualReview": { "status": "scan-compared-chapter10", "evidence": "book05-source/answer-0098.png", "sha256": "f4b78ada8adaf1177e5d3ede0855447dfe40c85b28fac76cef8c689af50abfb8", "nextChapterHeadingPDF": 210 } }, { "source": "answers-05", "pdfPage": 99, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "555272f565e37598d85d24f1daa14c6f860217e8d1ebadd0e587fa6e6bc3d4b2", "textHash": "4bfa5ce111427e16d3ef5fdd5633ec219ede696b071f997517329036b916dbd8", "characters": 1824, "flags": [], "candidateHeadings": [ "Exercises for Section 11.2" ], "candidateMarkers": [ { "id": "11.2.1", "label": null, "offset": 103 }, { "id": "11.2.2", "label": null, "offset": 1425 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0099.png", "sha256": "1f492623bdf0d5e6b0049ecf02c7b6d39616ef76d14a6ee59c26dfa90a8d0b49", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 100, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9c8ba7f6504dde27de3e7c1d69d7a6b02f469f8d17ceb8935036ed3eabfd431d", "textHash": "86c278b6670ec12de4694fad049802e4424f2020b6ab2443093cc6d721042146", "characters": 1936, "flags": [], "candidateHeadings": [ "Exercises for Section 11.3", "Exercises for Section 11.4" ], "candidateMarkers": [ { "id": "11.5.1", "label": null, "offset": 41 }, { "id": "11.3.2", "label": null, "offset": 1404 }, { "id": "11.4.1", "label": null, "offset": 1838 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0100.png", "sha256": "4ce77e9f781d6c2b1b02773825c2b9d436b4ad2be350ba2483cbb072fe29d7e0", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 101, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "e199194df57bd984e8b8d25c6fe316c56e30222b97a59aed5fa0219f7ced84dc", "textHash": "c0056a777e804752634e0a943a9daf44bdeead5068944539cc8e1c6b2ba41cfc", "characters": 1553, "flags": [], "candidateHeadings": [ "Exercises for Section 11.5" ], "candidateMarkers": [ { "id": "11.5.1", "label": null, "offset": 966 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0101.png", "sha256": "9b01276455f0ca84b92122ba177dbc4bc5c37d1ddd833785c33ece6fcfdf4edd", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 102, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "442ee9fc79514f498a99f8189877ba22d9dd53b54f5ab371d02b43036885ef3c", "textHash": "77094a7c534c8e6bf9184ceec03c191aa37a7a9a24dfcad048cad349c43e5278", "characters": 10551, "flags": [ "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "11.5.2", "label": null, "offset": 490 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0102.png", "sha256": "327ff3a1efaa13e64b15dc14938f528ec6b9413465be2fccc118fb5e7572b453", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 103, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "8caebff827f7a8f3ad92d138dbdb6dbbf0ebf6f57c2a9ebeb29ffb6d4ba1b277", "textHash": "12017a651853eaf56a9fe26b288d408d6bafb2d77b6d90e1fb83473a0c000312", "characters": 1167, "flags": [], "candidateHeadings": [ "Review Problems", "11.19 We don't need to convert a single fraction. We just count the ones with prime factorizations that include any prime other than 2 or 5: 3, 6, 7, and 9, so \\(\\frac{4}{4}\\) of them.", "11.20 We expand the factorials and then convert to decimal form:", "11.21 First, we use long division or another method to find that \\(\\frac{1}{37} = 0.027\\). We are looking for the 291\\(^{st}\\) digit in the 3-digit repeating block 0-2-7. Since 291 is a multiple of 3, we want the last digit in the trio, which is \\(\\underline{7}\\).", "Challenge Problems" ], "candidateMarkers": [ { "id": "11.18", "label": null, "offset": 51 }, { "id": "11.19", "label": null, "offset": 212 }, { "id": "11.20", "label": null, "offset": 397 }, { "id": "11.21", "label": null, "offset": 559 }, { "id": "11.22", "label": null, "offset": 824 }, { "id": "11.23", "label": null, "offset": 1107 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0103.png", "sha256": "a434fe6dc05908fdf03d047161b9cb4c77bd9043593edcc7fed7bcd3d3e6d201", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 104, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "0fa40258d5bae42327a6a12f1580db39c7a018f4e656e7910feff4d4375fd690", "textHash": "1f51c91427b6efd7f3ea7693fcf2fd97f69a9530bcb6e73a0f08f02f8ed7edbb", "characters": 2050, "flags": [], "candidateHeadings": [ "Challenge Problems", "11.26 First, we find the repeating decimal expansion of our fraction: \\(\\frac{1}{17} = 0.0588235294117647\\). We want the \\(4037^\\text{th}\\) digit of a 16-digit block of digits. The remainder from \\(4037 \\div 16\\) is 5, so our answer is the \\(5^\\text{th}\\) digit of the block, which is \\(\\boxed{2}\\).", "11.27 The first term is 1/100 and the common ratio is 7/100, so", "11.28 During the long division process, there are only p - 1 remainders after division by p at each step (digit). Once a remainder repeats itself during long division, the entire long division process repeats from that point forward, so the cycle of digits can be at most \\(\\boxed{p - 1}\\) digits long." ], "candidateMarkers": [ { "id": "11.24", "label": null, "offset": 309 }, { "id": "11.25", "label": null, "offset": 817 }, { "id": "11.26", "label": null, "offset": 964 }, { "id": "11.27", "label": null, "offset": 1265 }, { "id": "11.28", "label": null, "offset": 1743 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0104.png", "sha256": "088806a7400c813f39554467343b792439decab72af79d21331f427a1f0b13dd", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 105, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "e198fd3abb9c78cdd7abd51b47b2159c1ad2d3c0b00ae657c0d9c6db5663a19a", "textHash": "66e4227505391a413e203116449d1fe1356ac669574a3a1b03f3d2b8a1565cd2", "characters": 2925, "flags": [], "candidateHeadings": [ "11.30 Creating a geometric series whose denominators are powers of 10 helps us see that computing the first few digits is just a matter of computing a few powers of 2:", "11.32 If a number has a terminating decimal in base 8 then we can multiply it by \\(8^k = 2^{3k}\\) for some positive integer k and the result will be a base-8 integer. That is, the result will be integral. This means we can express the number as a fraction whose denominator is a \\(\\boxed{pure power of 2}\\)." ], "candidateMarkers": [ { "id": "11.29", "label": null, "offset": 34 }, { "id": "11.30", "label": null, "offset": 1408 }, { "id": "11.31", "label": null, "offset": 1788 }, { "id": "11.32", "label": null, "offset": 2164 } ], "visualReview": { "status": "scan-compared-chapter11", "evidence": "book05-source/answer-0105.png", "sha256": "419200b6e4d41cd37a8df54de87212f27417bbdece7ae79f114c6bee0686e764", "nextChapterHeadingPDF": 232 } }, { "source": "answers-05", "pdfPage": 106, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "27c4b21c4d36a53c6236e4e8f1f25187bcfd368fc245ecf77f37a51d1f8fb093", "textHash": "a1cd7c5df3c9ef320ffe23856b1bcd169ae2a4c93988d1b8aa26c2321979b2e9", "characters": 1506, "flags": [], "candidateHeadings": [ "Exercises for Section 12.2" ], "candidateMarkers": [ { "id": "12.2.1", "label": null, "offset": 74 }, { "id": "12.2.2", "label": null, "offset": 154 }, { "id": "12.2.3", "label": null, "offset": 931 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0106.png", "sha256": "83ee6b1ff35bb22d7d81cf859e107a35404737a80cb8c8210b0f4df41749a88e", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 107, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "e3af1478e0b33898fd8702ecf1cfa51d2e2855053d2da27a6801cbc871360b47", "textHash": "d07855e8bdf679a4d23e0956a37a6a77073e8263bf654053f5e8aabf7b8acd69", "characters": 2058, "flags": [], "candidateHeadings": [ "Exercises for Section 12.3" ], "candidateMarkers": [ { "id": "12.2.4", "label": null, "offset": 47 }, { "id": "12.2.5", "label": null, "offset": 602 }, { "id": "12.2.6", "label": null, "offset": 1256 }, { "id": "12.3.1", "label": null, "offset": 1740 }, { "id": "12.3.2", "label": null, "offset": 1872 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0107.png", "sha256": "0cccb5f1ad9cf9370032779b0d965a54cbb259bdb916e4f6a67d20953ad89c3c", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 108, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2a200e85df25c5dea497fc18380e35a7749d060a66a6fe2cf153f26d098ef0fb", "textHash": "c3b8e10e69548641aa4b561771a8e7b050e619d73bbe009d8da8689a5887ac85", "characters": 1512, "flags": [], "candidateHeadings": [ "Exercises for Section 12.4" ], "candidateMarkers": [ { "id": "12.3.3", "label": null, "offset": 13 }, { "id": "12.3.4", "label": null, "offset": 245 }, { "id": "12.3.5", "label": null, "offset": 576 }, { "id": "12.4.1", "label": null, "offset": 945 }, { "id": "12.4.2", "label": null, "offset": 1355 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0108.png", "sha256": "e1efa1086808345cfec6ef4c03ed3a5e1cb475339ac5dc63d371c54fee4159fd", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 109, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ec602f416a7f94b7cf5cf8b1b37305080a9d535682edbc21f9dce8f59447d648", "textHash": "68ffc519d484ffbe0184761fd5d9b4e1dd3bd6f85e30613d4fb183b0551077d6", "characters": 1854, "flags": [], "candidateHeadings": [ "Exercises for Section 12.5" ], "candidateMarkers": [ { "id": "12.4.3", "label": null, "offset": 47 }, { "id": "12.4.4", "label": null, "offset": 653 }, { "id": "12.4.5", "label": null, "offset": 874 }, { "id": "12.5.1", "label": null, "offset": 1198 }, { "id": "12.5.2", "label": null, "offset": 1489 }, { "id": "12.5.3", "label": null, "offset": 1640 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0109.png", "sha256": "32cbc9389222dad16d60f38af8d07e83158dde1dec5b7ccf9c91fc5d10fce81c", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 110, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "21fad7bf704937dc03e9ec4127ef3110ef24de6b680a1190d77f535ecae7bc8a", "textHash": "5936e4549c6b4afc500c5ac33526c283f84059d166becd49f19cd0bb8226b423", "characters": 2508, "flags": [], "candidateHeadings": [ "Exercises for Section 12.6" ], "candidateMarkers": [ { "id": "12.5.4", "label": null, "offset": 13 }, { "id": "12.5.5", "label": null, "offset": 217 }, { "id": "12.5.6", "label": null, "offset": 726 }, { "id": "12.6.1", "label": null, "offset": 1047 }, { "id": "12.6.2", "label": null, "offset": 2085 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0110.png", "sha256": "b65b40197f1a72e9d85a95fff4c215cd47c76eec48d9bfe505468e5c5c9ecb5a", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 111, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "3d4be1a405c4946bee938b5fecb9efd69fce56e97ede57c13dcdc4aced7f79a9", "textHash": "9e93f61bf9a4e355e548d28cde5b6356d5e660680518f02071792d550e90f59b", "characters": 2460, "flags": [], "candidateHeadings": [], "candidateMarkers": [ { "id": "12.6.3", "label": null, "offset": 362 }, { "id": "12.6.4", "label": null, "offset": 707 }, { "id": "12.6.5", "label": null, "offset": 1218 }, { "id": "12.6.6", "label": null, "offset": 2093 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0111.png", "sha256": "e85fe586709da44bbf89ffaba1d84d9a4c65df0f2d465c3033a523f2da86df9a", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 112, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "857d9c65830eb3c0720b8e9e80ec8370d75eeec20784aa63524605e66a67bad2", "textHash": "5eea2dcfb0c1b1a7a0da359686fc54c94760fb954b86e9f9bc810926057929ea", "characters": 1917, "flags": [], "candidateHeadings": [ "Review Problems", "Review Problems", "12.25 Integers congruent to 5 (mod 7) are the ones that can be written as \\( 7n + 5 \\) for some integer n. We are counting the values of n such that" ], "candidateMarkers": [ { "id": "12.23", "label": null, "offset": 33 }, { "id": "12.24", "label": null, "offset": 681 }, { "id": "12.25", "label": null, "offset": 1504 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0112.png", "sha256": "192a6d98e0e965e9f8460e281c9ec999308a14cd326e0f43e1b539e498503b0e", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 113, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "3cd4b5f9406ac8fa429754dc1f8fe1e802e15637f921ab2e4c48dee6aa877030", "textHash": "703cffc4cbee5df93c67f68335258578662f63eb56ac4cc58eebe1c0a0077233", "characters": 2200, "flags": [], "candidateHeadings": [ "12.26 If \\( n \\equiv -n \\pmod{12} \\), then \\( n - (-n) = 12k \\) for some integer \\( k \\).", "12.27 Let the number of jelly beans in the individual jars be \\( a \\), \\( b \\), \\( c \\), and \\( d \\). The modulo-12 residues of \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are 3, 5, 7, and 11. So,", "12.29 Multiplying \\( a \\equiv 19 \\pmod{30} \\) by 3, we get \\( 3a \\equiv 57 \\equiv 27 \\pmod{30} \\). This means that", "12.30 It takes a while to find that \\( 7^{10} \\equiv 1 \\pmod{11} \\). We use it to break down the problem:", "12.31 Note that \\( 9 \\equiv 2 \\pmod{7} \\) and \\( 5 \\equiv -2 \\pmod{7} \\):", "12.32 Squaring each of the modulo-6 residues, we find that perfect squares can only have modulo-6 residues of 0, 1, 3, and 4. As Susie already knows, this means that no perfect square leaves a remainder of 2 when divided by 6." ], "candidateMarkers": [ { "id": "12.26", "label": null, "offset": 47 }, { "id": "12.27", "label": null, "offset": 505 }, { "id": "12.28", "label": null, "offset": 858 }, { "id": "12.29", "label": null, "offset": 1229 }, { "id": "12.30", "label": null, "offset": 1565 }, { "id": "12.31", "label": null, "offset": 1802 }, { "id": "12.32", "label": null, "offset": 1968 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0113.png", "sha256": "a0c83e6337280a31ef9cd8b7737b5b5a9eaf8359bd1807bcbf17d2130add945a", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 114, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ac82a8fdf6f68b35799e0cf494ecc2c0e65ecbbc5546b952037ff1663249137e", "textHash": "daff889213faa03aea8f4a6b7950c5e595d5dbf6a499f1939c6fdc88946340f8", "characters": 6806, "flags": [ "truncated" ], "candidateHeadings": [ "Challenge Problems", "Challenge Problems", "12.33 The last two digits of a positive integer are its modulo-100 residue.", "12.34 One solution involves directly applying a formula for the sum of the first \\(n\\) positive integers (or for a more general arithmetic series). Our solution groups integers into pairs with constants sums (in the spirit of the derivation of formulas for arithmetic series). These pairings allow us to easily compute the modulo-200 residue:", "12.35 The difference between 617n and 943n is a multiple of 18, so", "12.36 Let's take a look at the modulo-8 residues of the first few summands:" ], "candidateMarkers": [ { "id": "12.33", "label": null, "offset": 39 }, { "id": "12.34", "label": null, "offset": 309 }, { "id": "12.35", "label": null, "offset": 779 }, { "id": "12.36", "label": null, "offset": 1006 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0114.png", "sha256": "34ba5e6903605ad3cbc4f56ae23184e5a75658d41eb57173443dcacf79d9da23", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 115, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "6d14b27580c9092352103e1b2f127a4f929cf25df947eb3d5b9b0f354863e111", "textHash": "9d410fbb7dd84ea9c7678aff04a6e75addcc33bf167d68b181b8ac0abac5fe75", "characters": 2891, "flags": [], "candidateHeadings": [ "12.39 From the units digit of n, we know that \\(n \\equiv 10 \\pmod{6}\\). Since 6 | 12, we also know that \\(n \\equiv 10 \\equiv 4 \\pmod{6}\\). Squaring, we get \\(n^2 \\equiv 4^2 \\equiv 16 \\equiv 4 \\pmod{6}\\).", "12.40 Since \\(69 \\equiv 90 \\equiv 125 \\pmod{N}\\), we know that both of the following are integers:", "12.41 Let n be the positive integer in question.", "12.42 This problem looks rather daunting. Without a clear direction, we might as well work out the remainder when the first term is divided by 7. If we are lucky, we’ll notice something along the way that will help us.", "12.43 We prove more generally that for k people, the number of people who shook hands with an odd number of others must be even.", "12.44 The sum of two integers is a multiple of 7 if and only if the sum of their modulo-7 residues is a multiple of 7. In order to get a grip on how we can maximize a subset without such a pair whose sum is a multiple of 7, we divide the integers from 1 to 50 inclusive into modulo-7 residue classes:" ], "candidateMarkers": [ { "id": "12.39", "label": null, "offset": 252 }, { "id": "12.40", "label": null, "offset": 457 }, { "id": "12.41", "label": null, "offset": 724 }, { "id": "12.42", "label": null, "offset": 1243 }, { "id": "12.43", "label": null, "offset": 2047 }, { "id": "12.44", "label": null, "offset": 2585 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0115.png", "sha256": "ea44a3ff52353fde2d829bf0fcc43445f77430c7a63f132149e96aac3e4b1154", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 116, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "729c15c5b74dc422627e68f5581335989e1554fa0dcc15bcc3d74b4ff73f2a35", "textHash": "cf877fa50321fbaffed94506386470bba941e8aca0ae7107cef44fa9b29ba389", "characters": 3379, "flags": [], "candidateHeadings": [ "Challenge Problems", "12.45 The first thing we do is compute modulo-5 residues for small Fibonacci numbers. Our hope is to find something helpful:" ], "candidateMarkers": [ { "id": "12.45", "label": null, "offset": 1024 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0116.png", "sha256": "104d660462baa0070db94bae27a4600efd2404babec7f237e62a209c545e68b7", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 117, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "39cbc5f474ce1588fa5bac60e2a34eee806f88ddb51a6103ebfe22c6205ab1cb", "textHash": "805dbedf9c1e8159b2a09a8982428370d7dff2a6ffee74ae0966ffda23cd5ce9", "characters": 2382, "flags": [], "candidateHeadings": [ "12.46 All primes greater than 2 are odd:", "12.47 The pigeonhole principle tells us that among any 51 integers, there are 2 with the same modulo-50 residue. Each of these integers can be written as \\( 50n + r \\) where \\( n \\) is an integer and \\( r \\) is the shared residue. Note that", "12.48 When applying modular arithmetic to difficult number theory problems, the difficult part is often coming up with the right modulus to use. The number of possible residues of squares in modulus 3, 4, and 8 are limited, which is often helpful for solving problems. In this case, modulo 4 can help us." ], "candidateMarkers": [ { "id": "12.46", "label": null, "offset": 401 }, { "id": "12.47", "label": null, "offset": 1192 }, { "id": "12.48", "label": null, "offset": 1745 } ], "visualReview": { "status": "scan-compared-chapter12", "evidence": "book05-source/answer-0117.png", "sha256": "b8dfdc33f499ab9f744330727b9f7aa372f806d9b2fc72ecd5ce6b22ec5a165f", "nextChapterHeadingPDF": 262 } }, { "source": "answers-05", "pdfPage": 118, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "657a50489aceb12daf617c988b81deb1466f41405b2f66fe629c4c7b28438e96", "textHash": "8d3732dac585d351178541a9f2a3b5de485d08ccf68ca8fec2d8b706e553d8d6", "characters": 2486, "flags": [], "candidateHeadings": [ "Exercises for Section 13.2" ], "candidateMarkers": [ { "id": "13.2.1", "label": null, "offset": 58 }, { "id": "13.2.2", "label": null, "offset": 2251 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0118.png", "sha256": "48a50d2d384a8f86f38ca8b65837c00b4343107a06aea914ba40b0ba4166689f", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 119, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "ed78eab521be1857de68259acc65555e7be5d70a329784a676cac56c4aaac0e2", "textHash": "87455b0a5270d40037398e77403d81834c94b31b1349fc5d5c4e7e167cc21f59", "characters": 1361, "flags": [], "candidateHeadings": [ "Exercises for Section 13.3" ], "candidateMarkers": [ { "id": "13.2.3", "label": null, "offset": 197 }, { "id": "13.2.4", "label": null, "offset": 795 }, { "id": "13.3.1", "label": null, "offset": 1122 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0119.png", "sha256": "5b8a13c3b414abd9d46a0371acac7b198db958ef6e48d65f1b692464db31fe47", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 120, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "10e3f799c5a869b4af6cff94269f325521263bd226623e66bf41fae14232c488", "textHash": "626a84347bc59a8b02e0d3b39ee969d95d5e1499546779cb0fb64429d70ccc05", "characters": 15236, "flags": [ "truncated" ], "candidateHeadings": [], "candidateMarkers": [ { "id": "13.3.2", "label": null, "offset": 13 }, { "id": "13.3.3", "label": null, "offset": 556 }, { "id": "13.3.4", "label": null, "offset": 1011 }, { "id": "13.3.5", "label": null, "offset": 1406 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0120.png", "sha256": "1147b2acc80df9bac506082248082293bba1065d2609de5b7f2e1efbbe695155", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 121, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "71637c69316106f8440642c9b6fdddc724091393e6a667ccacae471f5d632eb2", "textHash": "242316b3d4900f361ba1cd9729af306aac299e587229d39d9759220ef700dfb0", "characters": 2767, "flags": [], "candidateHeadings": [ "Review Problems", "13.11 An integer is a multiple of 9 if and only if the sum of its digits is a multiple of 9.", "13.12 An integer is a multiple of 11 if and only if the alternating sum of its digits is a multiple of 11." ], "candidateMarkers": [ { "id": "13.3.6", "label": null, "offset": 423 }, { "id": "13.3.7", "label": null, "offset": 1003 }, { "id": "13.11", "label": null, "offset": 1971 }, { "id": "13.12", "label": null, "offset": 2492 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0121.png", "sha256": "cf52ab361d7585e8d48bed440536a5619df16d4cd0f5d4bfae2820d798907536", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 122, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "7a9259dc42da1f5e145d5ae31a4a672d709e04de005276a52f984bb0a2b0ec4c", "textHash": "493cc7319364d6cb0d3f60cf398517ecbff28e27b1078436d5742ccc0d589a3c", "characters": 2453, "flags": [], "candidateHeadings": [ "Challenge Problems", "13.13 An integer is a multiple of 8 if and only if its last three digits form a multiple of 8.", "13.14 Since 72 = 8 \\(\\cdot\\) 9 and gcd(8, 9) = 1, we use the divisibility rules for 8 and 9 to find the digits A and B.", "13.15 Since 45 = 5 \\(\\cdot\\) 9 and gcd(5, 9) = 1, we use the divisibility rules for 5 and 9 to find the digits A and B.", "Challenge Problems", "13.16 We know the digits of the integer, so we think to apply any rule that comes to mind that uses all the digits, regardless of their order. Hence, we apply the divisibility rule for 9 and note that the nine-digit number is congruent to" ], "candidateMarkers": [ { "id": "13.13", "label": null, "offset": 240 }, { "id": "13.14", "label": null, "offset": 759 }, { "id": "13.15", "label": null, "offset": 1633 }, { "id": "13.16", "label": null, "offset": 2146 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0122.png", "sha256": "b375eea7bcd53cf62919f98104792eb4fb4b3c92083ab313a7e5fad8004a7938", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 123, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "64d4b50ad9f8923e1f85ea719e4f536a7417fbb68aa52b778dd907b930c58e73", "textHash": "4c29ae1ccc9527056b77aed70d38d310b9936089edde6091449bbb3dad5c4e04", "characters": 2776, "flags": [], "candidateHeadings": [ "13.17 An integer is a multiple of 4 if and only if its last two digits are a multiple of 4. Also, the units digit must be even in order for an integer to be even, so this gives us a good starting point for testing cases.", "13.18 A four-digit palindrome must be in the form ABBA for some digits A and B. Such a palindrome is a multiple of 9 if and only if the sum of its digits is a multiple of 9. In other words, \\(2(A + B)\\) must be a multiple of 9. Since 2 and 9 are relatively prime, \\(A + B\\) must be a multiple of 9.", "13.19 Since \\(99 = 9 \\cdot 11\\), and \\(gcd(9, 11) = 1\\), there is a temptation to try to build a divisibility rule by combining those for 9 and 11. However, noting that \\(100 = 10^2 \\equiv 1 \\pmod{99}\\) gives us a nicer possibility." ], "candidateMarkers": [ { "id": "13.17", "label": null, "offset": 172 }, { "id": "13.18", "label": null, "offset": 1059 }, { "id": "13.19", "label": null, "offset": 1715 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0123.png", "sha256": "a31da1ad023822ca85f2353173dd2565b689a12bf1f65c9f3dde917f87b47e48", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 124, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f172b6ca435f3275ee2bc747bfcbec16d3e865d1f04f9373b9b5f32b2bf36463", "textHash": "1b9905620d59cfddb10b15dd58f83e4a5cd16d689282fd78aee3d5251f52a38c", "characters": 2935, "flags": [], "candidateHeadings": [ "Challenge Problems", "13.20 Let's get a grip on the arithmetic behind the divisibility rule. For any given integer n, we can rewrite n as \\(10A + B\\), where B is the units digit of n. Now, let's be very precise about writing the meaning behind the divisibility rule mathematically. We are told that", "13.21 There are lots of three-digit positive integers with digit sums of 13, and they aren't easily described in a useful way. So, we'll start with the fact that the integers we're counting are congruent to 1 (mod 4). We want integers whose last two digits are congruent to 1 (mod 4). Such integers end in 01, 05, 09, 13, 17, ..., 97. Each of these 100/4 = 25 pairs of digits can be associated with at most one hundreds digit such that the sum of all three digits is 13. Since the hundreds digit is between 1 and 9 inclusive, we want the pairs of last two digits to have a sum between 4 and 12 inclusive. We throw out 01, 21, 49, 69, 77, 85, 89, and 97. This leaves 25 - 8 = 17 pairs of last two digits that can each be grouped with a hundreds digit to make a digit sum of 13.", "13.22 We are looking for an integer that is a multiple of lcm[7, 8, 9] = 504. Let our six-digit number be 523ABC for some digits A, B, and C. First we note that 523ABC is a multiple of 504 if and only if 523ABC - 504000 = 19ABC is a multiple of 504. Now, we need to find an integer 19ABC that is a multiple of 504. We could apply divisibility rules to the problem, but we can also simply go hunting! What makes hunting easier is the fact that 19000/504 is approximately equal to 19000/500 = 38. We note that 38 \\(\\cdot\\) 504 = 19152 and 39 \\(\\cdot\\) 504 = 19656. From these, we find that the six-digit numbers 523152 and 523656 are multiples of 504. The moral of the story is to keep an open mind as to when to use different methods to solve a problem.", "13.23 Let's take a look at the three simplest examples:" ], "candidateMarkers": [ { "id": "13.20", "label": null, "offset": 182 }, { "id": "13.21", "label": null, "offset": 1059 }, { "id": "13.22", "label": null, "offset": 1837 }, { "id": "13.23", "label": null, "offset": 2591 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0124.png", "sha256": "51c75c336cd6f71f9db1eb9c2a462084cd1c45ad6fec211099c2f0a5dbeef331", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 125, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "359be2de140007e6d1ac41d4a6dfa244f976528051101606078fdaba57ec4c9d", "textHash": "c92e5c2d62eae18b70fa62fec867fbf35f58faf710b23589e897f359d729ddce", "characters": 3072, "flags": [], "candidateHeadings": [ "13.24 Let the base-b digits of n be (from right to left) \\(d_0\\), \\(d_1\\), \\(d_2\\), ... Since \\(b \\equiv 1 \\pmod{b - 1}\\), we have", "13.25 First, we note that when n = 2, \\(n^3 - n = 6\\). This means that k | 6. With this in mind, we check to see if \\(n^3 - n\\) is always a multiple of 2 and 3. There are several possible ways to go about this, but we take a factoring approach:", "13.26 Let A, B, and C be digits of the five-digit palindrome ABCBA that is a multiple of 99. Thus" ], "candidateMarkers": [ { "id": "13.24", "label": null, "offset": 553 }, { "id": "13.25", "label": null, "offset": 1138 }, { "id": "13.26", "label": null, "offset": 1837 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0125.png", "sha256": "102081a2c82888d9c92479b2e21033dc25e3305c7c8c809b1bd12adce47bbb13", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 126, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "938da19408a6f791f9a438684f4b0aff3806885080baf8b3bde34d04413641f1", "textHash": "33c101459fb71d387b191d686873bb51ff11b8173ed44fa09ddbb1235125235d", "characters": 2938, "flags": [], "candidateHeadings": [ "Challenge Problems", "13.27 Let \\(d_0, d_1, d_2, \\ldots\\) be the base-12 digits of \\(n\\) so that", "13.28 The last four digits of an integer determine its modulo-\\(2^4\\) residue. This motivates us to examine possible integers modulo 16. This seems reasonable since any power of 2 with at least 4 digits is a multiple of 16.", "13.29 Since \\(15 = 3 \\cdot 5\\), and \\(gcd(3, 5) = 1\\), we use the divisibility rules for 3 and 5 to help us find \\(n\\). Since \\(5 \\mid n\\), \\(n\\) has a units digit of either 0 or 5. However, we are told that all of \\(n\\)'s digits are either 0 or 8, so the units digit of \\(n\\) is 0. Now we note that \\(3 \\mid n\\), so the sum of \\(n\\)'s digits is a multiple of 3. However, the sum of \\(n\\)'s digits is also the number of 8's that are digits of \\(n\\). This smallest total of 8's is 3, where \\(3 \\cdot 8 = 24\\) is a multiple of 3. Finally, we want the smallest possible value of \\(n\\), so we include no more digits and \\(n = 8880\\) is the smallest positive multiple of 15 all of whose digits are 0 or 8. The answer to the problem is \\(n/15 = 8880/15 = \\underline{592}\\).", "13.30 We could solve all parts of this problem by dividing each of the four integers into longer strings of 1's until no remainder is left:" ], "candidateMarkers": [ { "id": "13.27", "label": null, "offset": 19 }, { "id": "13.28", "label": null, "offset": 737 }, { "id": "13.29", "label": null, "offset": 1436 }, { "id": "13.30", "label": null, "offset": 2205 } ], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0126.png", "sha256": "1288eaab19f7884cddb8e5a366ee4070593862017da123ecd9d83c30182e921b", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 127, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "b183ea27d726b220f762c7c9fc2425c5866417f1658af0c2a1074e734626b6b2", "textHash": "d68b0e7f0711148da59dcd809571805e57a9a46e2552e08aa5fbb65c1995d12a", "characters": 2294, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-chapter13", "evidence": "book05-source/answer-0127.png", "sha256": "b4af608f0a995a96004ad8d9e3bc2f79a41a0a8a1261ef3a74530dd109c63b72", "nextChapterHeadingPDF": 276 } }, { "source": "answers-05", "pdfPage": 128, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "4935da242e4c043df9c8ef6437f5ec139ef707d87023cfeeeac79b468aeaffc1", "textHash": "51799c2ea7bb376fce8540213a36c34824a4b90d48dd16e982f96dda1bbfae64", "characters": 1315, "flags": [], "candidateHeadings": [ "Exercises for Section 14.2" ], "candidateMarkers": [ { "id": "14.2.1", "label": null, "offset": 58 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0128.png", "sha256": "564c3c93ec07faab174d104df01e6fe6f9ba998b6d9ef0865d89c3c896a9f74d", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 129, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9a5ac1d53d619913267431da50712340991b3010aed0fced229e9ae8f57984c5", "textHash": "d51d67c6452abc86d55d2975828674d77fa4ad545318d139c657ba75c61cdab3", "characters": 1949, "flags": [], "candidateHeadings": [ "1.1 ≡ 1 (mod 11) ⇒ 1⁻¹ ≡ 1 (mod 11)", "2.6 ≡ 1 (mod 11) ⇒ 2⁻¹ ≡ 6 (mod 11)", "3.4 ≡ 1 (mod 11) ⇒ 3⁻¹ ≡ 4 (mod 11)", "4.3 ≡ 1 (mod 11) ⇒ 4⁻¹ ≡ 3 (mod 11)", "5.9 ≡ 1 (mod 11) ⇒ 5⁻¹ ≡ 9 (mod 11)", "6.2 ≡ 1 (mod 11) ⇒ 6⁻¹ ≡ 2 (mod 11)", "7.8 ≡ 1 (mod 11) ⇒ 7⁻¹ ≡ 8 (mod 11)", "8.7 ≡ 1 (mod 11) ⇒ 8⁻¹ ≡ 7 (mod 11)", "9.5 ≡ 1 (mod 11) ⇒ 9⁻¹ ≡ 5 (mod 11)", "10.10 ≡ 1 (mod 11) ⇒ 10⁻¹ ≡ 10 (mod 11)", "Exercises for Section 14.3" ], "candidateMarkers": [ { "id": "14.2.2", "label": null, "offset": 31 }, { "id": "1.1", "label": null, "offset": 39 }, { "id": "2.6", "label": null, "offset": 77 }, { "id": "3.4", "label": null, "offset": 115 }, { "id": "4.3", "label": null, "offset": 153 }, { "id": "5.9", "label": null, "offset": 191 }, { "id": "6.2", "label": null, "offset": 229 }, { "id": "7.8", "label": null, "offset": 267 }, { "id": "8.7", "label": null, "offset": 305 }, { "id": "9.5", "label": null, "offset": 343 }, { "id": "10.10", "label": null, "offset": 381 }, { "id": "14.2.3", "label": null, "offset": 423 }, { "id": "14.2.4", "label": null, "offset": 600 }, { "id": "14.3.1", "label": null, "offset": 1419 }, { "id": "14.3.2", "label": null, "offset": 1599 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0129.png", "sha256": "4c5053dd23e3406e3ee01423b51a3a5e1a95ec9c017175a240669a2a1e7f2a52", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 130, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a9dafaac654b1c127ec287e22e9c8b632911f2e4883fadb9d801f64d87f97fed", "textHash": "d97aa7c17a35c7a7dce8c575be2524b4ebc0b218f7b4b7670a9f63540cd01a72", "characters": 1692, "flags": [], "candidateHeadings": [ "Exercises for Section 14.4" ], "candidateMarkers": [ { "id": "14.3.3", "label": null, "offset": 65 }, { "id": "14.3.4", "label": null, "offset": 370 }, { "id": "14.4.1", "label": null, "offset": 655 }, { "id": "14.4.2", "label": null, "offset": 990 }, { "id": "14.4.3", "label": null, "offset": 1400 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0130.png", "sha256": "5667a9859057f2dc6513e83b3e33c8baf1ea3984e3b98922ce503b5c5152e8a0", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 131, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "0b297e3aa0deff615cd8e30e76ed433851edad747ff4cae673c851bb04f798df", "textHash": "aacdede0a8b74895c12831b6ca9b080ccfc31d486c8c78236f5e5a1c61d4e7f2", "characters": 1525, "flags": [], "candidateHeadings": [ "Review Problems" ], "candidateMarkers": [ { "id": "14.4.4", "label": null, "offset": 220 }, { "id": "14.4.5", "label": null, "offset": 882 }, { "id": "14.20", "label": null, "offset": 1413 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0131.png", "sha256": "ab5dd9045804f89b7c381cc590be9dd8000bf4b9b8d5601d91ecd57857bdfdfa", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 132, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "61d9e157123cab3fcaac767ce25f1100dfb4fc60c8ab97dc09c27550420450cd", "textHash": "2086752dd6ca1172acd4e83a82b912e564d1e7b0f960dc8c55d37968920863f9", "characters": 1492, "flags": [], "candidateHeadings": [ "Review Problems", "14.21 Modulo-20 residues with no inverses are the ones not relatively prime to 20. Those are 0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, and 18.", "14.22 Since the product of m - 1 with itself is congruent to 1 (mod m), m - 1 is its own inverse:", "14.24 We begin by solving each of the linear congruences in the system to get" ], "candidateMarkers": [ { "id": "14.21", "label": null, "offset": 324 }, { "id": "14.22", "label": null, "offset": 464 }, { "id": "14.23", "label": null, "offset": 602 }, { "id": "14.24", "label": null, "offset": 1125 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0132.png", "sha256": "ad8360109cb5c549318181c56a8f43c195d37a14ab8e6c1e3de9e470fbb15e50", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 133, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "491fef6f46c6869f46892264c4fad876575bf341e7fa3a4f0cd631a0e2098a79", "textHash": "4bfb30f805587f2f9e9f4ce41f8d4aa30438a8bc0db398dab876ece9600f7115", "characters": 1724, "flags": [], "candidateHeadings": [ "14.25 From the first two congruences, we know that", "14.26 We are looking for integers \\(n\\) such that \\(200 \\leq n \\leq 500\\) and", "14.27 We are looking for the smallest positive integer \\(x\\) that satisfies the system of linear congruences" ], "candidateMarkers": [ { "id": "14.25", "label": null, "offset": 83 }, { "id": "14.26", "label": null, "offset": 768 }, { "id": "14.27", "label": null, "offset": 1461 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0133.png", "sha256": "39375763a5745405ae33c0525241b522828b59f8f89cda7754a94fa13365f6c9", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 134, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a97c96dcf62e5d83c1bbdd1d4f40961c419edb96fbff96f8741d63a9956e27d5", "textHash": "70f2a7f8b974ec5e4a3ae8f8abdd1dcb0102156f68baf14a166bf7e7ee54edd8", "characters": 1824, "flags": [], "candidateHeadings": [ "Challenge Problems", "Challenge Problems", "14.28 We need to identify integers congruent to 7 (mod 11) and 10 (mod 12). Call such an integer n. Then we know that" ], "candidateMarkers": [ { "id": "14.28", "label": null, "offset": 837 }, { "id": "14.29", "label": null, "offset": 1334 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0134.png", "sha256": "2680a8f4260fdb3ebfd147cb5e00c2c107202e772cc9e93a89a8b4d3e965d061", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 135, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "c9a79c921b354b49375e157b57ee42342c3325380f1ffad8a0b11dc58e90310a", "textHash": "6b871f1a4694e8182a7b1209d3747961aab6f8031299bb5c27f9891ea3e79664", "characters": 3135, "flags": [], "candidateHeadings": [ "14.30 We want to identify mutual days of rest so that we can count them. Days of rest for Adam are in the form 4a for some positive integer a. Days of rest for Ben are in the form 10b - 2, 10b - 1, or 10b where b is a positive integer. We want days when Adam's and Ben's time off overlap. We want solutions, n, that satisfy any one of the three following systems of linear congruences:", "14.31 Let A be the number of four-digit integers that leave a remainder of 2 when divided by 7. Let B be the number of four-digit integers that leave a remainder of 4 when divided by 5. Let C be the number of four-digit integers that both leave a remainder of 2 when divided by 7 and a remainder of 4 when divided by 5. Our goal is to compute \\((A - C) + (B - C)\\) (make sure you see why).", "14.32 The product of two modulo 120 units digits has a units digit of 1 if the units digits are modulo 120 inverses of each other. Only integers relatively prime to 120 have modulo 120 inverses. We count" ], "candidateMarkers": [ { "id": "14.30", "label": null, "offset": 615 }, { "id": "14.31", "label": null, "offset": 1475 }, { "id": "14.32", "label": null, "offset": 2926 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0135.png", "sha256": "033582b07e4cc228be1ea336de37a03e6b8ad8f9a2fe2cecc795c3c747559994", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 136, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a9c103d377eb920a7eeac567e1d6ddf2b80ed0f4b96752f1e8fb0ffc8c88b0b9", "textHash": "fc0fa29dd252d4d9fc01b4d5943c47bc11d2a905a820a648451d937e5fc3084d", "characters": 1187, "flags": [], "candidateHeadings": [ "Challenge Problems", "14.33 A 50-digit prime is not a multiple of 2, 3, or 5. Thus, we know that one of the following two systems of linear congruences holds:" ], "candidateMarkers": [ { "id": "14.33", "label": null, "offset": 581 } ], "visualReview": { "status": "scan-compared-chapter14", "evidence": "book05-source/answer-0136.png", "sha256": "7a6d5910821827f5bd04f9a04889161ef2327de195c5235db072abbfebb5829e", "nextChapterHeadingPDF": 298 } }, { "source": "answers-05", "pdfPage": 137, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "1b338b678c0aadb3091c1dc334bba455f41f0683472cb7fbae13311dfc076f6d", "textHash": "6f9ab4a5a30e84a2d1a1eac1243ec7b1593f5ce48d2293125b9de4b1264f9423", "characters": 1653, "flags": [], "candidateHeadings": [ "Review Problems", "15.20 We need to compute", "15.21 We are told that the sum of two primes is equal to a third, so for prime numbers \\(a, b\\), and \\(c\\),", "15.22 The smallest such integer is 2457. If our number were 4 or more times another number, then it would be at least 4.2457 = 9828, which is too big. So our number is 2 or 3 times another number. Also note that the smaller number (the number we are multiplying 2 or 3 by) must begin with 2, because 2.4257 = 8514 is too big." ], "candidateMarkers": [ { "id": "15.20", "label": null, "offset": 65 }, { "id": "15.21", "label": null, "offset": 631 }, { "id": "15.22", "label": null, "offset": 1093 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0137.png", "sha256": "8cd1345e7416323b2b1475fe39737eeab6ba21ea325892cec53c6bc7bec656dd", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 138, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "efed265b7400c28ca886b12949775739ca9ae540f76d41de1ed77b12f93f66c2", "textHash": "ad5815e28eba54779be127744687adc002fa406cdae1a79dcd0edcdb9aca8b44", "characters": 2174, "flags": [], "candidateHeadings": [ "Review Problems", "15.23 Since each of the two products includes 1212, we can factor to make things easier:", "15.24 All the numbers in this problem are powers of 2, so we can change everything into forms of powers of 2 in order to get the clearest look at the equation:", "15.25 We can start applying divisibility rules for each of the smallest primes in order, using our understanding of modular arithmetic to help.", "15.26 Since the product \\(8^{12} \\cdot 25^8\\) is a multiple of several powers of 10, it makes sense to reorganize the product in this way:", "15.27 We are looking for the smallest positive integer n such that" ], "candidateMarkers": [ { "id": "15.23", "label": null, "offset": 358 }, { "id": "15.24", "label": null, "offset": 511 }, { "id": "15.25", "label": null, "offset": 827 }, { "id": "15.26", "label": null, "offset": 1273 }, { "id": "15.27", "label": null, "offset": 1967 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0138.png", "sha256": "05d7ebe9c32c994edb790e743c209ec94d59edf9af265dcdf04d711369733e9a", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 139, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "2e1c3861b5f740488c928a68aac52371b1c20119109eae83df91f10e2ae4aa05", "textHash": "26f289822d071edc14334c673ac95a1a67040b7749d14c5fc554d695389297fd", "characters": 2375, "flags": [], "candidateHeadings": [ "15.28 Once again, we apply parity to a sum of primes. Since the sum of the three primes is even, one of them must be the even prime, 2. This means the sum of the other two primes is 22. Our possible choices for the remaining prime numbers are 3 and 19, 5 and 17, and 11 and 11. There are six possible orderings of (2, 3, 19), six possible orderings of (2, 5, 17), and three possible orderings of (2, 11, 11), for a total of 15 ordered triples.", "15.29 Solving an equation based on this problem would be tough, but we can make casework relatively easy by narrowing the range of integers whose cubes have four-digits. Since \\(10^3 = 1000\\) and \\(22^3 = 10648\\), there are only twelve 4-digit numbers which are cubes at all. We want the largest four-digit number equal to the cube of the sum of its digits, so we can work downwards from cubes of possible digit sums:", "15.30 We see powers of 61 multiplied by 1, -3, 3, and -1. These remind us of the coefficients of \\((x - 1)^3\\):", "15.31 Let's call the number we are looking for N. For each m = 2, 3, ..., 10 we have" ], "candidateMarkers": [ { "id": "15.28", "label": null, "offset": 793 }, { "id": "15.29", "label": null, "offset": 1238 }, { "id": "15.30", "label": null, "offset": 1993 }, { "id": "15.31", "label": null, "offset": 2246 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0139.png", "sha256": "432b770db53b41aafe62490deb69ea37f75e028083ec9492f6a336d5afcf3a45", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 140, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "9b48b0b49ea2b66abbc248eb9fbf0112987e615954679a677723ca67cf08a577", "textHash": "a6268d44666f255614d86b7473688477364fa0ebf3f211f1be43185001462338", "characters": 2991, "flags": [], "candidateHeadings": [ "Review Problems", "15.32 While it's usually easiest to find the GCD of a pair of integers using prime factorization, there is a nice relationship between the integers in this problem: \\(39654 = 39600 + 54 = 50 \\cdot 792 + 54\\). This allows an easy application of the Euclidean algorithm:", "15.33 If A and B were both odd, then A + B would be the even prime 2. But 2 is less than the sum of any primes, so either A or B must be even and therefore 2. Since A - B is prime, A > B, so B = 2.", "15.34 Let a, b, and c be integers such that \\(0 < a < b < c\\) and \\(a^2 + b^2 + c^2 = 125\\). Since", "15.35 We see powers of 3, so we change the integers to exponential form for easier computation:", "15.36 We can use the relationship between \\(2^{2006}\\), \\(5^{2006}\\), and the fact that the decimal system is based on powers of 10. We note that \\(2^{2006} \\cdot 5^{2006} = 10^{2006}\\) is a 2007-digit number. We can write the powers of 2 and 5 in a way that makes the last step clear:" ], "candidateMarkers": [ { "id": "15.32", "label": null, "offset": 356 }, { "id": "15.33", "label": null, "offset": 727 }, { "id": "15.34", "label": null, "offset": 1258 }, { "id": "15.35", "label": null, "offset": 2162 }, { "id": "15.36", "label": null, "offset": 2345 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0140.png", "sha256": "94d147da1bd4c899fc07b9db7b9ac4292f1929f6350ddf28293d009da336ae68", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 141, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "c0af80a5f502fc89f497f02ea1614055f150e05bdcf07cb917027611003c7836", "textHash": "dc1fe03c04a059f8791039425b6d1189815b6bc0c710d325a1d6199bcca04b6b", "characters": 2914, "flags": [], "candidateHeadings": [ "15.37 If \\(n\\) is one less than a multiple of each of 3, 5, 7, 9, and 11, \\(n + 1\\) must be a multiple of all of them, and therefore a multiple of their LCM. Since 3 is a divisor of 9, we can ignore it in the calculation. Since 5, 7, 9, and 11 are all coprime in pairs, the LCM is", "15.38 We recall that any number ending in 2 or 4 must be divisible by 2, and any number ending in 5 must be divisible by 5. Since two prime numbers are formed, one of them must end in one of those three numbers. Since a prime can't be divisible by 2 unless it is equal to 2 or by 5 unless it is equal to 5, one of the primes must be either 2 or 5, and the other prime must end in 7. If one of the primes is 2, the other prime can either be 457 or 547. If one of the primes is 5, the other can either be 247 or 427. 247 is not prime, and the smallest of 2 \\(\\cdot\\) 457, 2 \\(\\cdot\\) 547, and 5 \\(\\cdot\\) 427 is 2 \\(\\cdot\\) 457 \\(\\cdot\\) \\(\\underline{914}\\).", "15.39 Factoring thus difference of squares, we have", "15.40 There are not very many two-digit numbers that are 5 more than a multiple of 7, so we can just list them out:", "15.41 We can use what we know about the value of the square to narrow down the possibilities. We start with broad estimates: \\(300^2 = 90,000 < 152AB1 < 160,000 = 400^2\\), so the first digit of the square root is 3. Indeed, \\(380^2 = 144,400 < 152AB1\\), so the first two digits of the square root are either 38 or 39. Since the last digit of the number 152AB1 is 1, its square root must have a last digit of either 1 or 9. Since 152AB1 is close to midway between 144,400 and 160,000, the two possibilities we must check for its square root are 389 and 391. \\(389^2 = 151,321 < 152AB1\\), but \\(391^2 = 152881\\), which is of the required form, so the answer is \\(3 + 9 + 1 = \\underline{13}\\).", "15.42 Seeing all those powers of 5, we rewrite the whole expression in terms of 5:" ], "candidateMarkers": [ { "id": "15.37", "label": null, "offset": 395 }, { "id": "15.38", "label": null, "offset": 880 }, { "id": "15.39", "label": null, "offset": 1538 }, { "id": "15.40", "label": null, "offset": 1681 }, { "id": "15.41", "label": null, "offset": 1966 }, { "id": "15.42", "label": null, "offset": 2658 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0141.png", "sha256": "1567cc15e18f4d3916afc5fc63915de0da593bb3d031c43ad45402a039953114", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 142, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "36633157718c275e1d04b817cced08d5575014ad57463cff7a7276bf9c2622de", "textHash": "d39ef0d134b700e356f96edf4be2274b1c8549d0c0b2802724881af696d32673", "characters": 2842, "flags": [], "candidateHeadings": [ "Review Problems", "15.43 When an integer n is 1 more than a multiple of each 2 and 3, then n - 1 is a multiple of both 2 and 3 and therefore a multiple of 6. We write \\(n = 6m + 1\\) for a nonnegative integer m. Since \\(0 < n < 100\\),", "15.44 Since the sum of the two prime numbers is 20, they can be expressed as \\(10 + x\\) and 10 - x for some integer x. We can compute the product as a difference of squares:", "15.45 Notice that \\((10)(6) = 2 \\cdot (5)(6)\\). The squares of two numbers and two times their product remind us of the factorization \\((x - y)^2 = x^2 - 2xy + y^2\\). When x = 5 and y = 6, we get the expression we are trying to evaluate:", "15.46 We recall that a number is a multiple of 5 if and only if its units digit is 0 or 5. Therefore, none of the given 2, 4, or 8 can be the last digit. We order 2, 4, and 8 from least to greatest to minimize the integer, so the smallest such number will be of the form 248x. The smallest possibility is \\(\\boxed{2480}\\).", "15.47 We factor \\(66666 = 3 \\cdot 22222\\). Then \\(A = 66666^4 = (3 \\cdot 22222)^4 = 3^4 \\cdot 22222^4 = 81B\\). Dividing by B we get \\(A/B = \\boxed{81}\\).", "15.48 Since \\(2^4 = 16\\), we can use a difference of squares twice:", "15.49 The temptation is to write out all of the first 25 primes, look at all of their remainders when divided by 4, and multiply, but there is a much simpler way. Since 2 is a prime, the product of the first 25 primes is going to be a multiple of 2, so its remainder when divided by 4 will be either 2 or 0. But this product is not a multiple of 4, since no other primes have a factor of 2, so the remainder cannot be 0. Therefore, the remainder is \\(\\boxed{2}\\)." ], "candidateMarkers": [ { "id": "15.43", "label": null, "offset": 16 }, { "id": "15.44", "label": null, "offset": 519 }, { "id": "15.45", "label": null, "offset": 981 }, { "id": "15.46", "label": null, "offset": 1270 }, { "id": "15.47", "label": null, "offset": 1594 }, { "id": "15.48", "label": null, "offset": 1749 }, { "id": "15.49", "label": null, "offset": 2373 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0142.png", "sha256": "a720e94658a2ac085438c73cc8b7bb6af8e9199cdf7fd8b8b381e2f643d82c72", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 143, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "a2ab7f2783076fc4c7b80fce840f7b1aa0b980ab91d1425560964c62a9224732", "textHash": "678a6233ba7cdc3c60e9a8cd8cdfd2cbc753a312d421706a241797e9d26d3750", "characters": 1968, "flags": [], "candidateHeadings": [ "15.50 Rather than exponentiating and subtracting, we factor \\(2^{19}\\) out of each term, getting", "15.51 Let N > 1 be an integer that, when divided by each of 2, 3, ..., 10 and 11, leaves a remainder of 1.", "15.52 The repeated digits give us a clue about a common factor. We write \\(8008=8\\cdot1001\\), and \\(14014=14\\cdot1001\\). Since, gcd(8, 14) = 2,", "15.53 Any multiple of \\(9^{3}\\) can be written in the form \\(9^{3}n\\) for some integer n. Our goal is to count the number of integers in this form between \\(9^{4}\\) and \\(9^{6}\\). We divide the inequality", "15.54 Those coefficients look familiar (a row of Pascal's triangle!), and the fact that the powers of 53 decrease while those of 3 increase gives it away: the given expression is the expansion of \\((53-3)^{5}\\). Now the computation is relatively simple:", "15.55 We can use both factorization and the trick for squaring integers with units digits of 5." ], "candidateMarkers": [ { "id": "15.50", "label": null, "offset": 25 }, { "id": "15.51", "label": null, "offset": 180 }, { "id": "15.52", "label": null, "offset": 581 }, { "id": "15.53", "label": null, "offset": 797 }, { "id": "15.54", "label": null, "offset": 1151 }, { "id": "15.55", "label": null, "offset": 1436 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0143.png", "sha256": "8709d7861b237383ae1c4e9c4dccda08de1df29cd2a3f118ad7ea6e91fea6285", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 144, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "67c5218478bd726e707c6ca287b20b8cdb20da61d800eda6628dd59ad9d0d486", "textHash": "dfaccd5fae0d620ec898af2ea30acf4c651bd5d9d1ca2e24028df2dcb502ef87", "characters": 1867, "flags": [], "candidateHeadings": [ "Review Problems", "15.56 Since each denominator is at least 1, we can determine the value of each variable in succession by subtracting out the largest positive integer less than or equal to the right-hand side of the equation.", "15.57 There are a couple of nice approaches to this problem. We could start by factoring the difference of cubes, but since the numbers being used are constructed from such easy-to-multiply integers, we can work the problem algebraically.", "15.58 We can cancel \\(2^{2000}\\) from every part of the fraction for easy computation:", "15.59 Seeing the powers of 18 makes us think about a factorization of the cube of a sum. However, the “coefficients” don’t match. We notice powers of 2 added – particularly the last term, \\(2^3\\). Indeed, we can compute this as the cube of a sum:", "15.60 This will be easier to evaluate if we change everything to decimal form:" ], "candidateMarkers": [ { "id": "15.56", "label": null, "offset": 16 }, { "id": "15.57", "label": null, "offset": 649 }, { "id": "15.58", "label": null, "offset": 1177 }, { "id": "15.59", "label": null, "offset": 1368 }, { "id": "15.60", "label": null, "offset": 1751 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0144.png", "sha256": "edeac6164f8c9459f15fe5bb771f30459b5d4808288403e578572240605d173f", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 145, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "f76aac65e7f8f6a2e8d4600819204f4d030bf92f3955a99f1624d3893c02c2b5", "textHash": "13a96e407a09ef66fedef1b593a76bd132a6e9dcaa5d945025bff0f537dceaf7", "characters": 1219, "flags": [], "candidateHeadings": [ "15.61 The fraction \\(\\frac{n - 13}{5n + 6}\\) is reducible if and only if \\(\\gcd(5n + 6, n - 13) > 1\\). Using the Euclidean algorithm,", "15.62 The numbers are being listed such that an integer with a higher power of 3 in its prime factorization comes after an integer with a lower power of 3 in its prime factorization. This means the last number in the list will have the highest possible power of 3 in its prime factorization. We note that" ], "candidateMarkers": [ { "id": "15.61", "label": null, "offset": 241 }, { "id": "15.62", "label": null, "offset": 673 } ], "visualReview": { "status": "scan-compared-chapter15", "evidence": "book05-source/answer-0145.png", "sha256": "e261f9d4a404b368e62cb2fcb1d430553fd636816b4e0a4214c230e80e43048c", "nextChapterHeadingPDF": 318 } }, { "source": "answers-05", "pdfPage": 146, "sourcePDF": "K:\\prep2college\\AOPS全套23本\\5-9年纪初级+答案册\\05-Introduction to Number Theory-答案.pdf", "ocrHash": "aa7f723bfe258549dc9832600eb80b47d83d37bd8a620c24c70ed50684ee8bd6", "textHash": "c9a44d80bced9e1786cc42d8f569036ca699d321c5886dbaf73c93dffb778004", "characters": 1624, "flags": [], "candidateHeadings": [], "candidateMarkers": [], "visualReview": { "status": "scan-compared-final-source-audit", "category": "publication-and-contents", "evidence": "book05-source/answer-0146.png", "sha256": "04e479e747a2f08715c7ef454ed41bdeb739e2fa9c139f0916135ae2f1c8009f" } } ], "sectionMapping": [ { "id": "1.1", "title": "Introduction", "candidateHeadingPages": [ 16 ], "sourcePdfPages": [ 16, 17 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.2", "title": "Making Integers Out of Integers", "candidateHeadingPages": [ 18 ], "sourcePdfPages": [ 18, 19, 20, 21 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.3", "title": "Integer Multiples", "candidateHeadingPages": [ 22 ], "sourcePdfPages": [ 22, 23, 24, 25, 26 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.4", "title": "Divisibility of Integers", "candidateHeadingPages": [ 26 ], "sourcePdfPages": [ 26, 27, 28, 29 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.5", "title": "Divisors", "candidateHeadingPages": [ 29 ], "sourcePdfPages": [ 29, 30, 31, 32, 33 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.6", "title": "Using Divisors", "candidateHeadingPages": [ 33 ], "sourcePdfPages": [ 33, 34 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.7", "title": "Mathematical Symbols", "candidateHeadingPages": [ 35 ], "sourcePdfPages": [ 35 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "1.8", "title": "Summary", "candidateHeadingPages": [ 36 ], "sourcePdfPages": [ 36, 37, 38 ], "sourceMappingStatus": "scan-confirmed" }, { "id": "2.1", "title": "Introduction", "candidateHeadingPages": [ 40 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "2.2", "title": "Primes and Composites", "candidateHeadingPages": [ 40 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "2.3", "title": "Identifying Primes I", "candidateHeadingPages": [ 43 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "2.4", "title": "Identifying Primes II", "candidateHeadingPages": [ 46 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "2.5", "title": "Summary", "candidateHeadingPages": [ 51 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.1", "title": "Introduction", "candidateHeadingPages": [ 54 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.2", "title": "Common Divisors", "candidateHeadingPages": [ 54 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.3", "title": "Greatest Common Divisors (GCDs)", "candidateHeadingPages": [ 56 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.4", "title": "Common Multiples", "candidateHeadingPages": [ 58 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.5", "title": "Remainders", "candidateHeadingPages": [ 61 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.6", "title": "Multiples, Divisors, and Arithmetic", "candidateHeadingPages": [ 65 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.7", "title": "The Euclidean Algorithm", "candidateHeadingPages": [ 68 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "3.8", "title": "Summary", "candidateHeadingPages": [ 75 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.1", "title": "Introduction", "candidateHeadingPages": [ 78 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.2", "title": "Factor Trees", "candidateHeadingPages": [ 79 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.3", "title": "Factorization and Multiples", "candidateHeadingPages": [ 83 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.4", "title": "Factorization and Divisors", "candidateHeadingPages": [ 87 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.5", "title": "Rational Numbers and Lowest Terms", "candidateHeadingPages": [ 90 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.6", "title": "Prime Factorization and Problem Solving", "candidateHeadingPages": [ 93 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.7", "title": "Relationships Between LCMs and GCDs", "candidateHeadingPages": [ 95 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "4.8", "title": "Summary", "candidateHeadingPages": [ 99 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "5.1", "title": "Introduction", "candidateHeadingPages": [ 106 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "5.2", "title": "Counting Divisors", "candidateHeadingPages": [ 106 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "5.3", "title": "Divisor Counting Problems", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "5.4", "title": "Divisor Products", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "5.5", "title": "Summary", "candidateHeadingPages": [ 119 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.1", "title": "Introduction", "candidateHeadingPages": [ 124 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.2", "title": "Some Special Primes", "candidateHeadingPages": [ 124 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.3", "title": "Factorials, Exponents and Divisibility", "candidateHeadingPages": [ 126 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.4", "title": "Perfect, Abundant, and Deficient Numbers", "candidateHeadingPages": [ 130 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.5", "title": "Palindromes", "candidateHeadingPages": [ 132 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "6.6", "title": "Summary", "candidateHeadingPages": [ 135 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "7.1", "title": "Introduction", "candidateHeadingPages": [ 140 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "7.2", "title": "Problems", "candidateHeadingPages": [ 140 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "7.3", "title": "Summary", "candidateHeadingPages": [ 152 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.1", "title": "Introduction", "candidateHeadingPages": [ 156 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.2", "title": "Counting in Bundles", "candidateHeadingPages": [ 156 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.3", "title": "Base Numbers", "candidateHeadingPages": [ 160 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.4", "title": "Base Number Digits", "candidateHeadingPages": [ 163 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.5", "title": "Converting Integers Between Bases", "candidateHeadingPages": [ 165 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.6", "title": "Unusual Base Number Problems", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "8.7", "title": "Summary", "candidateHeadingPages": [ 176 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.1", "title": "Introduction", "candidateHeadingPages": [ 180 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.2", "title": "Base Number Addition", "candidateHeadingPages": [ 180 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.3", "title": "Base Number Subtraction", "candidateHeadingPages": [ 183 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.4", "title": "Base Number Multiplication", "candidateHeadingPages": [ 185 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.5", "title": "Base Number Division and Divisibility", "candidateHeadingPages": [ 187 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "9.6", "title": "Summary", "candidateHeadingPages": [ 190 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "10.1", "title": "Introduction", "candidateHeadingPages": [ 192 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "10.2", "title": "Units Digits in Arithmetic", "candidateHeadingPages": [ 192 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "10.3", "title": "Base Number Units Digits", "candidateHeadingPages": [ 199 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "10.4", "title": "Unit Digits Everywhere!", "candidateHeadingPages": [ 202 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "10.5", "title": "Summary", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.1", "title": "Introduction", "candidateHeadingPages": [ 210 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.2", "title": "Terminating Decimals", "candidateHeadingPages": [ 210 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.3", "title": "Repeating Decimals", "candidateHeadingPages": [ 216 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.4", "title": "Converting Decimals to Fractions", "candidateHeadingPages": [ 220 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.5", "title": "Base Numbers and Decimal Equivalents", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "11.6", "title": "Summary", "candidateHeadingPages": [ 227 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.1", "title": "Introduction", "candidateHeadingPages": [ 232 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.2", "title": "Congruence", "candidateHeadingPages": [ 233 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.3", "title": "Residues", "candidateHeadingPages": [ 239 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.4", "title": "Addition and Subtraction", "candidateHeadingPages": [ 242 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.5", "title": "Multiplication and Exponentiation", "candidateHeadingPages": [ 247 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.6", "title": "Patterns and Exploration", "candidateHeadingPages": [ 253 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "12.7", "title": "Summary", "candidateHeadingPages": [ 257 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "13.1", "title": "Introduction", "candidateHeadingPages": [ 262 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "13.2", "title": "Divisibility Rules", "candidateHeadingPages": [ 262 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "13.3", "title": "Divisibility Rules With Algebra", "candidateHeadingPages": [], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "13.4", "title": "Summary", "candidateHeadingPages": [ 273 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "14.1", "title": "Introduction", "candidateHeadingPages": [ 276 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "14.2", "title": "Modular Inverses and Simple Linear Congruences", "candidateHeadingPages": [ 277 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "14.3", "title": "Solving Linear Congruences", "candidateHeadingPages": [ 282 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "14.4", "title": "Systems of Linear Congruences", "candidateHeadingPages": [ 287 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "14.5", "title": "Summary", "candidateHeadingPages": [ 293 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.1", "title": "Introduction", "candidateHeadingPages": [ 298 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.2", "title": "Familiar Factors and Divisibility", "candidateHeadingPages": [ 298 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.3", "title": "Algebraic Methods of Arithmetic", "candidateHeadingPages": [ 302 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.4", "title": "Useful Forms of Numbers", "candidateHeadingPages": [ 307 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.5", "title": "Simplicity", "candidateHeadingPages": [ 309 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" }, { "id": "15.6", "title": "Summary", "candidateHeadingPages": [ 312 ], "sourcePdfPages": null, "sourceMappingStatus": "candidate-only-needs-scan" } ], "sourceInventoryComplete": true, "recoveredTOCPage15": { "sourcePDFPage": 15, "evidence": "book05-source/page-0015.png", "evidenceHash": "b5f65d4dfbde4f9fb3c2160a853038cb4bf21af2b5673029d20f6b90b6f2bf46", "printedPages": { "12.1": 217, "12.2": 218, "12.3": 224, "12.4": 227, "12.5": 232, "12.6": 238, "12.7": 242, "13.1": 247, "13.2": 247, "13.3": 255, "13.4": 258, "14.1": 261, "14.2": 262, "14.3": 267, "14.4": 272, "14.5": 278, "15.1": 283, "15.2": 283, "15.3": 287, "15.4": 292, "15.5": 294, "15.6": 297 }, "status": "TOC-scan-confirmed; actual section boundaries pending scans" } }