{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0001","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e9e35ef519df11bf01204390d1425a70abe879b890ea0aab18236ac8cacdc8","family_id":"qf-mod32-totient-coprime-residues","seed":202611016001,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":1}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=1. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(1, [1], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (1, [1], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0002","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"827c4a25b48a87b22a32ec621c06a16eb5259c1df275745bb319e1bb739f507a","family_id":"qf-mod32-totient-coprime-residues","seed":202611016002,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":2}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=2. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(2, [1], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (2, [1], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0003","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9fab8bd0a53a2c832d0b65b5c39e478e481b252b52898c8434a197e56e92d2a","family_id":"qf-mod32-totient-coprime-residues","seed":202611016003,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":3}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=3. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(3, [1, 2], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (3, [1, 2], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0004","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f8bdd261eb9cb3a57f2d1e3dc32c4b605f7ac8c34374cc0e50d8a91b2331c50","family_id":"qf-mod32-totient-coprime-residues","seed":202611016004,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":4}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=4. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(4, [1, 3], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (4, [1, 3], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0005","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4471b05ebe7f3145c2fa70e08f36619bb172ca7af20640ed41e4e1f5ed9b5482","family_id":"qf-mod32-totient-coprime-residues","seed":202611016005,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":5}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=5. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(5, [1, 2, 3, 4], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (5, [1, 2, 3, 4], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0006","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090926b664b7c284b5d62e60afccf711953d58bf8fca90742b6e386b14bc131b","family_id":"qf-mod32-totient-coprime-residues","seed":202611016006,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":6}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=6. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(6, [1, 5], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (6, [1, 5], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0007","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e517b8bc7886423dad9fafee477ff4baf0044e8968265c383f4d764e84b6c67c","family_id":"qf-mod32-totient-coprime-residues","seed":202611016007,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":7}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=7. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(7, [1, 2, 3, 4, 5, 6], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (7, [1, 2, 3, 4, 5, 6], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0008","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5455138ff98b6f55170939c69ce1c809a33f7fea4e9ddef202db527fa8bd49b6","family_id":"qf-mod32-totient-coprime-residues","seed":202611016008,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":8}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=8. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(8, [1, 3, 5, 7], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (8, [1, 3, 5, 7], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0009","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d814a3e339b95e31f485b4b33c15907948ffe6cbaa04899375ad5ef48ef8464a","family_id":"qf-mod32-totient-coprime-residues","seed":202611016009,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":9}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=9. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(9, [1, 2, 4, 5, 7, 8], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (9, [1, 2, 4, 5, 7, 8], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0010","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a69bed639d425ece69167a03ccd76658ad36a5252395623d23cd6f62544b2f9d","family_id":"qf-mod32-totient-coprime-residues","seed":202611016010,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":10}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=10. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(10, [1, 3, 7, 9], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (10, [1, 3, 7, 9], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0011","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba50b6cb396dc9b0bd32e133699d3d8164cb6a088703231c18720e1192856971","family_id":"qf-mod32-totient-coprime-residues","seed":202611016011,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":11}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=11. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0012","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a372cdc249be8443a769d02a5284e40f684737dca12159c7e0d106ecea29bfbe","family_id":"qf-mod32-totient-coprime-residues","seed":202611016012,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":12}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=12. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(12, [1, 5, 7, 11], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (12, [1, 5, 7, 11], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0013","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c031ec05ea1a4b2e953b6b82d9d17b13b703dc8ae97bc12e32eebf4b4dab8c02","family_id":"qf-mod32-totient-coprime-residues","seed":202611016013,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":13}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=13. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0014","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e54a0c4001a4a3831368e30006af929a91628e9d6235ebdf496190b0c35a2e0f","family_id":"qf-mod32-totient-coprime-residues","seed":202611016014,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":14}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=14. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(14, [1, 3, 5, 9, 11, 13], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (14, [1, 3, 5, 9, 11, 13], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0015","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec34c35aa8b18ca9deb70aac7df0144e1885af41f912677216fad8dc7f218542","family_id":"qf-mod32-totient-coprime-residues","seed":202611016015,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":15}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=15. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0016","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4caf587f743f3239db09ccbeee96d6cbcac7f439db506ff525d1fe5d0e2192de","family_id":"qf-mod32-totient-coprime-residues","seed":202611016016,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":16}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=16. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0017","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d2538419ce10fab96fe059ba0155db18b1f7d9c9b34ee42e44288d7eb515c8d","family_id":"qf-mod32-totient-coprime-residues","seed":202611016017,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":17}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=17. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0018","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adcb096ed037ca12d4b9379e7d8ee4303364ebda2b0195e0f750ad52b0167ad2","family_id":"qf-mod32-totient-coprime-residues","seed":202611016018,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":18}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=18. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(18, [1, 5, 7, 11, 13, 17], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (18, [1, 5, 7, 11, 13, 17], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0019","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48e93b0ee58e4ec895b6405d61076dcc8988bf839f66564f7387a57767486052","family_id":"qf-mod32-totient-coprime-residues","seed":202611016019,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":19}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=19. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0020","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7f468c5ea71158d4d9c438a1c1144aa4e30c574a2085b39f06d9ae05d257db7","family_id":"qf-mod32-totient-coprime-residues","seed":202611016020,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":20}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=20. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0021","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ae25be84c9ee839b549a1fa3cbbc6ff37dd86cab19ba19262869f167455cb86","family_id":"qf-mod32-totient-coprime-residues","seed":202611016021,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":21}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=21. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0022","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"535caf1e09fc4f5c0d93fffa931a9e32e314b430625727a8e8f9444c22e123bd","family_id":"qf-mod32-totient-coprime-residues","seed":202611016022,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":22}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=22. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0023","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7904b3a4073c6e7a46b0ebee7f0c9b978ba691fe4671f8b2da2ffd7482679f11","family_id":"qf-mod32-totient-coprime-residues","seed":202611016023,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":23}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=23. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0024","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2c7dcbfb73bf4ffcfa3531640de2695ceefbb23f2c3d5e77c76ca88625d09ac","family_id":"qf-mod32-totient-coprime-residues","seed":202611016024,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":24}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=24. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0025","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13e4aef7ce6c56d145037b444a4cdb42855b2a4cb9463a7c5fe847cb0c53d495","family_id":"qf-mod32-totient-coprime-residues","seed":202611016025,"semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":25}"},"fields":{"title":"Use Every Representative from 1 through n, Including the n=1 Convention","stem":"Positive integer n=25. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.","answer":{"kind":"expression","value":"(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n)","explanation":"Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n)."},"skills":["Use Every Representative from 1 through n, Including the n=1 Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0026","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca11fe24cb9e34668bdaf7875a114614e24adf79d11b3b105cade9c0564f882c","family_id":"qf-mod32-totient-value","seed":202611016026,"semantic_template":"qf-mod32-totient-value/{\"n\":1}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=1, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(1, [1], 1)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (1, [1], 1)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0027","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d6c505a4b122b01b0298a47205bfbc7edca83dbfd6814929c8671dc1869d28a","family_id":"qf-mod32-totient-value","seed":202611016027,"semantic_template":"qf-mod32-totient-value/{\"n\":2}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=2, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(2, [1], 1)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (2, [1], 1)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0028","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b820d7f5951e7d94849b1fc2d030994755dba8f0a1fb651f1343247ff9acdd8e","family_id":"qf-mod32-totient-value","seed":202611016028,"semantic_template":"qf-mod32-totient-value/{\"n\":3}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=3, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(3, [1, 2], 2)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (3, [1, 2], 2)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0029","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e70b57020ed50203ca90e28e1e5b14f230029eb429295741e28e585bbdcbc83","family_id":"qf-mod32-totient-value","seed":202611016029,"semantic_template":"qf-mod32-totient-value/{\"n\":4}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=4, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(4, [1, 3], 2)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (4, [1, 3], 2)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0030","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"104a4a1cdcb0837c215d6aca21c7d252f32dd475f641b2a7e153fb832bddfb7c","family_id":"qf-mod32-totient-value","seed":202611016030,"semantic_template":"qf-mod32-totient-value/{\"n\":5}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=5, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(5, [1, 2, 3, 4], 4)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (5, [1, 2, 3, 4], 4)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0031","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d06752f9e332e591b489e1411e85248368be8ca364907d2b9fbe04d4bf3a690d","family_id":"qf-mod32-totient-value","seed":202611016031,"semantic_template":"qf-mod32-totient-value/{\"n\":6}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=6, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(6, [1, 5], 2)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (6, [1, 5], 2)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0032","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5617aaadc7d733c43a322bda8d23b17afc68f9fe7b4f07684b85774ad14e0575","family_id":"qf-mod32-totient-value","seed":202611016032,"semantic_template":"qf-mod32-totient-value/{\"n\":7}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=7, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(7, [1, 2, 3, 4, 5, 6], 6)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (7, [1, 2, 3, 4, 5, 6], 6)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0033","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48b2faf55a784bdd856f24860ab3e95bdf7861d90984fbd4886175fe6fffa5b6","family_id":"qf-mod32-totient-value","seed":202611016033,"semantic_template":"qf-mod32-totient-value/{\"n\":8}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=8, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(8, [1, 3, 5, 7], 4)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (8, [1, 3, 5, 7], 4)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0034","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2ee7c8e1f1394e66d956d82189a219467bd2cb7e0f1985bb41f212f54c55574","family_id":"qf-mod32-totient-value","seed":202611016034,"semantic_template":"qf-mod32-totient-value/{\"n\":9}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=9, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(9, [1, 2, 4, 5, 7, 8], 6)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (9, [1, 2, 4, 5, 7, 8], 6)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0035","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02ccb5769402727f40d6e9b7a1e5f3618a7b728eee59b24ab24075cf30960e7","family_id":"qf-mod32-totient-value","seed":202611016035,"semantic_template":"qf-mod32-totient-value/{\"n\":10}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=10, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(10, [1, 3, 7, 9], 4)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (10, [1, 3, 7, 9], 4)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0036","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb5ce351fe4ea8b2aaf5ed213752561ff1da63de22bb7bc05b50d86336d1708f","family_id":"qf-mod32-totient-value","seed":202611016036,"semantic_template":"qf-mod32-totient-value/{\"n\":11}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=11, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0037","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75e40412ce267b45fea50629656f243b6f9031b3b66df315cac9f01478960e03","family_id":"qf-mod32-totient-value","seed":202611016037,"semantic_template":"qf-mod32-totient-value/{\"n\":12}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=12, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(12, [1, 5, 7, 11], 4)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (12, [1, 5, 7, 11], 4)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0038","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3f816e95a6492d62f2b41bdedb196939b3d4b6917c7f2580cbf637c14c9f41","family_id":"qf-mod32-totient-value","seed":202611016038,"semantic_template":"qf-mod32-totient-value/{\"n\":13}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=13, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0039","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7460ce432b0499bdf3969a0c3a37ed176715e807c0a0955b2cff788f554f087e","family_id":"qf-mod32-totient-value","seed":202611016039,"semantic_template":"qf-mod32-totient-value/{\"n\":14}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=14, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(14, [1, 3, 5, 9, 11, 13], 6)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (14, [1, 3, 5, 9, 11, 13], 6)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0040","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14ece22065e8ba1552f6518cf9f65ce791ac84082a397d2824e8d7f044d8284a","family_id":"qf-mod32-totient-value","seed":202611016040,"semantic_template":"qf-mod32-totient-value/{\"n\":15}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=15, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(15, [1, 2, 4, 7, 8, 11, 13, 14], 8)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (15, [1, 2, 4, 7, 8, 11, 13, 14], 8)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0041","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b928215e126b9a6acd5fda79f4325cfe89f49a2d4e12f040a1ac5720e3df8e51","family_id":"qf-mod32-totient-value","seed":202611016041,"semantic_template":"qf-mod32-totient-value/{\"n\":16}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=16, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(16, [1, 3, 5, 7, 9, 11, 13, 15], 8)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (16, [1, 3, 5, 7, 9, 11, 13, 15], 8)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0042","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83669fa1e60adf3d93c3041af461d37cd4d0dc5644b8f001fa82d2e4fed7b7d","family_id":"qf-mod32-totient-value","seed":202611016042,"semantic_template":"qf-mod32-totient-value/{\"n\":17}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=17, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0043","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a72f2fa30abb9bb7e6c0653abaceba288e24415edd60117058f26bacc20df2b2","family_id":"qf-mod32-totient-value","seed":202611016043,"semantic_template":"qf-mod32-totient-value/{\"n\":18}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=18, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(18, [1, 5, 7, 11, 13, 17], 6)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (18, [1, 5, 7, 11, 13, 17], 6)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0044","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d165f09fd158bf6ba522575e79a02ad603d84fe296f70f04df7c9c1600ffbbe","family_id":"qf-mod32-totient-value","seed":202611016044,"semantic_template":"qf-mod32-totient-value/{\"n\":19}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=19, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0045","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4be641fc970708aba94a93568eb3d8a3c91fc71834bbc1e3d26dcda3ec280ff","family_id":"qf-mod32-totient-value","seed":202611016045,"semantic_template":"qf-mod32-totient-value/{\"n\":20}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=20, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(20, [1, 3, 7, 9, 11, 13, 17, 19], 8)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (20, [1, 3, 7, 9, 11, 13, 17, 19], 8)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0046","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b1fbae6d051d4e0c159e3977f8ed91f155484d199724437f6364d07f18863cc","family_id":"qf-mod32-totient-value","seed":202611016046,"semantic_template":"qf-mod32-totient-value/{\"n\":21}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=21, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0047","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bd5ae2ebc88100add66f67d46317e9e514e13841097a450c4fc393fe73a7395","family_id":"qf-mod32-totient-value","seed":202611016047,"semantic_template":"qf-mod32-totient-value/{\"n\":22}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=22, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0048","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536233ee78a1c154be7d806c6cd3e4b0b41130e9be51f77ef81759938f532e3a","family_id":"qf-mod32-totient-value","seed":202611016048,"semantic_template":"qf-mod32-totient-value/{\"n\":23}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=23, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0049","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaf3f712de467f73907f946d5651f16e9f11b546898e5500f49c3a120348df07","family_id":"qf-mod32-totient-value","seed":202611016049,"semantic_template":"qf-mod32-totient-value/{\"n\":24}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=24, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(24, [1, 5, 7, 11, 13, 17, 19, 23], 8)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (24, [1, 5, 7, 11, 13, 17, 19, 23], 8)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0050","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e7c488dac85e733b117752bb82a594147ffbeac277c55475233b170af67459c","family_id":"qf-mod32-totient-value","seed":202611016050,"semantic_template":"qf-mod32-totient-value/{\"n\":25}"},"fields":{"title":"Euler Totient Is the Exact Count of Coprime Representatives","stem":"Positive integer n=25, using representatives 1 through n. Give (n,coprime_residues,phi_n).","answer":{"kind":"expression","value":"(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20)","explanation":"List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20)."},"skills":["Euler Totient Is the Exact Count of Coprime Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0051","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"446d366e6892052aa49d18b5903eef9377d3af21850a947ea463340507f2f61f","family_id":"qf-mod32-totient-boundary-prime","seed":202611016051,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":1}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=1, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(1, 1, phi_1_equals_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (1, 1, phi_1_equals_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0052","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab8be6de53f5a9daa83ad9cc254d9c5b6f448c566c6ee4ac0e153071054521c4","family_id":"qf-mod32-totient-boundary-prime","seed":202611016052,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":2}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=2, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(2, 1, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (2, 1, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0053","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"600377d22cbae148731cdb2f2cf2d2bef87d99054bace52f3c1017e76a92298c","family_id":"qf-mod32-totient-boundary-prime","seed":202611016053,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":3}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=3, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(3, 2, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (3, 2, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0054","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"418ab19e065ca6fab11e158478c118ed54dcb4cf8bc35bbad6ff6a32d7616502","family_id":"qf-mod32-totient-boundary-prime","seed":202611016054,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":4}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=4, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(4, 2, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (4, 2, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0055","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda2d31fc08d0636193bb6e77fd784aff2a580c957a126cf33e0625318b56bac","family_id":"qf-mod32-totient-boundary-prime","seed":202611016055,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":5}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=5, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(5, 4, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (5, 4, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0056","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72b19628e87f7f7e01eb79aa7543ef9984d96d4c8fb2593fc4a07093a0f04057","family_id":"qf-mod32-totient-boundary-prime","seed":202611016056,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":6}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=6, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(6, 2, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (6, 2, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0057","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"052444aa216ad1e02836d14de9ff7a93bb2235a055c725ab4c8c03fecd3a7d9b","family_id":"qf-mod32-totient-boundary-prime","seed":202611016057,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":7}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=7, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(7, 6, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (7, 6, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0058","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b5edd71b6f9fc8a40a2328bb4ca2a4e2213add079dcd8ae72202f4ff9b1de31","family_id":"qf-mod32-totient-boundary-prime","seed":202611016058,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":8}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=8, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(8, 4, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (8, 4, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0059","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b17801d724126adb2bcb5c6628068c1cfd416c704f99fb82aa046806bc14f8ff","family_id":"qf-mod32-totient-boundary-prime","seed":202611016059,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":9}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=9, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(9, 6, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (9, 6, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0060","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce2eec87e5c9930cabbc3b5acfb20234a87b490b70d237659fcc4e9d7b54d1d","family_id":"qf-mod32-totient-boundary-prime","seed":202611016060,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":10}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=10, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(10, 4, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (10, 4, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0061","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ddd8548b1a1a348835a6f5deb3efdd5adddbaf20da69b253572cac6bd1a8a39","family_id":"qf-mod32-totient-boundary-prime","seed":202611016061,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":11}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=11, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(11, 10, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (11, 10, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0062","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"821e9b0d7565b485f5e4c4ce0c4629d1d492ef0b9438d078926c875ba2396e3a","family_id":"qf-mod32-totient-boundary-prime","seed":202611016062,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":12}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=12, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(12, 4, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (12, 4, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0063","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d9c2ace177ec414f2738667e4858762de1a39974eaddaa0378de1143db0a8a8","family_id":"qf-mod32-totient-boundary-prime","seed":202611016063,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":13}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=13, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(13, 12, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (13, 12, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0064","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2a7e518d69ceabca729a575a31aadd22c0964a20f16d7afa3adfc7d19799af3","family_id":"qf-mod32-totient-boundary-prime","seed":202611016064,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":14}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=14, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(14, 6, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (14, 6, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0065","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2cb532d35d30a288a4a7e6cd969e532a59ac5b16fcf54921a926fc0431c61dc","family_id":"qf-mod32-totient-boundary-prime","seed":202611016065,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":15}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=15, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(15, 8, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (15, 8, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0066","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8484997de2700fa2aaa0ac8da9e23ff90d1ab1f151f895bc3f933e5c3a16239d","family_id":"qf-mod32-totient-boundary-prime","seed":202611016066,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":16}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=16, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(16, 8, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (16, 8, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0067","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d85c5c6d0aeca13de9858d02477a82250d6adf0a0ab8f2348f01b8c4211a62d1","family_id":"qf-mod32-totient-boundary-prime","seed":202611016067,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":17}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=17, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(17, 16, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (17, 16, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0068","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd053272898a235d0d68226675f7ebc42bd847420cf3c2a885b3dbb31eea7c13","family_id":"qf-mod32-totient-boundary-prime","seed":202611016068,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":18}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=18, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(18, 6, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (18, 6, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0069","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb6dc5d6cf3919b6e767d96e729f506ad600e219c8c3041dbd8ac7e2a0871001","family_id":"qf-mod32-totient-boundary-prime","seed":202611016069,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":19}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=19, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(19, 18, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (19, 18, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0070","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60f229fd1ed902c6acee203cc775f3cef9872483e621f4506499b6698e995ef4","family_id":"qf-mod32-totient-boundary-prime","seed":202611016070,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":20}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=20, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(20, 8, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (20, 8, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0071","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ed29ec862ead009a67f6c2bfb7a3797ac0f4e3f986c55303bc9b31901506c4","family_id":"qf-mod32-totient-boundary-prime","seed":202611016071,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":21}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=21, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(21, 12, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (21, 12, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0072","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"369ee713225f5557ac25a0db8ab7388aa1c9577bb15b7725b694ef3723bfaedb","family_id":"qf-mod32-totient-boundary-prime","seed":202611016072,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":22}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=22, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(22, 10, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (22, 10, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0073","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ce6dd8066241652d0ea77587848a5993961cf5884bce0c0b01218cec27c88c7","family_id":"qf-mod32-totient-boundary-prime","seed":202611016073,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":23}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=23, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(23, 22, prime_phi_equals_n_minus_1)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (23, 22, prime_phi_equals_n_minus_1)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0074","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"771041ce9889dae86f86d245077cdf4be6870f39f90efaa4cb38c7db6deabd7b","family_id":"qf-mod32-totient-boundary-prime","seed":202611016074,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":24}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=24, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(24, 8, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (24, 8, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0075","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b805511c0b01ad6686892a3472b12000904c9b09f23d6d3be263068ddb41fe53","family_id":"qf-mod32-totient-boundary-prime","seed":202611016075,"semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":25}"},"fields":{"title":"Keep phi(1), the Prime Case and the Composite Boundary Distinct","stem":"Positive integer n=25, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.","answer":{"kind":"expression","value":"(25, 20, composite_no_prime_shortcut)","explanation":"Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (25, 20, composite_no_prime_shortcut)."},"skills":["Keep phi(1), the Prime Case and the Composite Boundary Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0076","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5397f2f398391d84b92f9b913f51ef3649af13ab0747b16eccb0e566cb20ed8","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016076,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":3}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=3 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 3, 1, 2, 2, 2, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 3, 1, 2, 2, 2, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0077","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17abf8abd535d175eb308302a3bd7cb3a25f7720d8eb47301e06951880f9cb34","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016077,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":5}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=5 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 5, 1, 4, 4, 4, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 5, 1, 4, 4, 4, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0078","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee19a1427cb0e882725fc1abc2f2e41e3a5d2927a9ed2165a2ab74f6fd58249","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016078,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":7}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=7 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 7, 1, 6, 6, 6, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 7, 1, 6, 6, 6, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0079","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4df9ef0231317af9150eb24b85dd9d8da76c595655f72e8b6aa3c302d2d44b3","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016079,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":9}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=9 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 9, 1, 6, 6, 6, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 9, 1, 6, 6, 6, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0080","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0912a061666c95346617ae069f296c6b057c2a4a9dda97fd0740a94d168024b","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016080,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":11}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=11 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0081","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6093d0b4c399c58ebe810c935687da43cd719f4497f178372d14cffa688403cc","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016081,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":13}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=13 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 13, 1, 12, 12, 12, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 13, 1, 12, 12, 12, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0082","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"018581d05eb9173973b97d025d47e663e0d226d178181586d5f91e67bdddbe3f","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016082,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":15}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=15 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 15, 1, 8, 8, 8, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 15, 1, 8, 8, 8, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0083","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8808df8c0b54ed9385c2f8701ccb2df78ce3553208f08da4bb7b2fdb3b693989","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016083,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":17}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=17 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 17, 1, 16, 16, 16, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 17, 1, 16, 16, 16, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0084","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7be81d407053a71ab41ce03b6d0bb140485378a8dca1a32bec1419c86e9fce65","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016084,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":19}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=19 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 19, 1, 18, 18, 18, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 19, 1, 18, 18, 18, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0085","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd77256b6553a975790ab7cb7e2a9f1c45ced03b6a2b9bc24c7af399159f1ab6","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016085,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":21}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=21 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 21, 1, 12, 12, 12, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 21, 1, 12, 12, 12, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0086","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5af959c88449e7d1fe1bc453701806e71f242963f2206c0fe6e6976b88f83c9","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016086,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":23}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=23 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 23, 1, 22, 22, 22, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 23, 1, 22, 22, 22, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0087","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1244c20a1ba32ea13010fbb6995067819479a1d102dc506bc03877620627a71","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016087,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":25}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=25 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 25, 1, 20, 20, 20, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 25, 1, 20, 20, 20, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0088","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5f37f8e2cbfdeb939753d331367c24ac7692ccd22a520c094831822bf295247","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016088,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":27}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=27 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 27, 1, 18, 18, 18, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 27, 1, 18, 18, 18, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0089","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93e91fc76c07060da337f5ceea43861e621d38ff270415f273d03964a05eca75","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016089,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":29}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=29 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 29, 1, 28, 28, 28, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 29, 1, 28, 28, 28, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0090","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3833ec21211fbf8c578ea580a6d88186d0561277c25e81bdf51339e051a97bc2","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016090,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":31}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=31 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 31, 1, 30, 30, 30, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 31, 1, 30, 30, 30, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0091","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b025f2500ba60f8c356740c168ce3b21a9e1e1470b6845e082cca00df2b7be57","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016091,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":33}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=33 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 33, 1, 20, 20, 20, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 33, 1, 20, 20, 20, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0092","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7de65f0fe26349d5e171d0fa7d460411fa295431089efaf48e993236a7535cc0","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016092,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":35}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=35 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 35, 1, 24, 24, 24, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 35, 1, 24, 24, 24, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0093","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0e9453b24cb1eab0e86953772619c60566afad333daaf2835335599b5c330a","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016093,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":37}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=37 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 37, 1, 36, 36, 36, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 37, 1, 36, 36, 36, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0094","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd4d83c2603db21a342accc2c1348c5dbb78ceee89560d50ad924357f367092c","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016094,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":39}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=39 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 39, 1, 24, 24, 24, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 39, 1, 24, 24, 24, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0095","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4eb21ea63aa8ebf62b5e767c89c4e5d88794947fc1b7be8f6d3e84e2c287d28a","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016095,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":41}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=41 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 41, 1, 40, 40, 40, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 41, 1, 40, 40, 40, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0096","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38089efbad4305952d075ccca20e27379a9a13667e5a35b314d62db2129345b0","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016096,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":43}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=43 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 43, 1, 42, 42, 42, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 43, 1, 42, 42, 42, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0097","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38e952f943896e60013f7ae39e37d4343c77f64acfbfc6696ea2e9b7f77b3433","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016097,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":45}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=45 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 45, 1, 24, 24, 24, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 45, 1, 24, 24, 24, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0098","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2205f8eb83f40eefcfa5388c528574b1c6bfa8c1d68e32d414c69fdaac37ba1","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016098,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":47}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=47 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 47, 1, 46, 46, 46, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 47, 1, 46, 46, 46, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0099","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd4a9768c65da4f66cc1d2f537124d2d9591fee84d1b75a11e328c1099787703","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016099,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":49}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=2, n=49 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(2, 49, 1, 42, 42, 42, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 49, 1, 42, 42, 42, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r06-euler-totient-function-0100","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be0a16659c4edc258859a7d51c2311855c1093f6f10389e8c614f8a3c1595d6d","family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016100,"semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":3,\"n\":4}"},"fields":{"title":"Totient Multiplicativity Requires the Coprimality Premise","stem":"Positive integers m=3, n=4 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).","answer":{"kind":"expression","value":"(3, 4, 2, 2, 4, 4, Yes, instance_only_with_coprimality)","explanation":"Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (3, 4, 2, 2, 4, 4, Yes, instance_only_with_coprimality)."},"skills":["Totient Multiplicativity Requires the Coprimality Premise"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}