{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r06-euler-theorem-number-theory-0001","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"051b1b881628aa0e6762cd9774857a72cd596f59ca175ab8ba286ff9973191c2","family_id":"qf-mod38-eulerthm-congruence","seed":202611076001,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":2,\"n\":5}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(5, 2, 4, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (5, 2, 4, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0002","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f7cebe8797bad9375b20279db09af7e14ec4c30f3d8edf4a1753a55d08b5b73","family_id":"qf-mod38-eulerthm-congruence","seed":202611076002,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":5,\"n\":6}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(6, 5, 2, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (6, 5, 2, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0003","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e8148fd88032a46ba0af8f19312ffcf42fff6138a69e50b2cbd326fa5420533","family_id":"qf-mod38-eulerthm-congruence","seed":202611076003,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":4,\"n\":7}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(7, 4, 6, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (7, 4, 6, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0004","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed6e57855dde931135ad20c630b7bef8c48e7a90ae4719d5c5e4b7b85bb99417","family_id":"qf-mod38-eulerthm-congruence","seed":202611076004,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":5,\"n\":8}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(8, 5, 4, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (8, 5, 4, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0005","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"692e456156eeebcc118db1da4740c8d88e8e85310b8864ddabff42de8e50e03c","family_id":"qf-mod38-eulerthm-congruence","seed":202611076005,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":7,\"n\":9}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(9, 7, 6, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (9, 7, 6, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0006","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da730110053d84d34a660f8a9192a09c3d77fe14a71a9e936b86bf8810ec4a8","family_id":"qf-mod38-eulerthm-congruence","seed":202611076006,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":7,\"n\":10}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(10, 7, 4, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (10, 7, 4, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0007","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a94f1a18699eea2e3dabaae0e41859477f0b85ae4173cbe98e4edaea7a5598e3","family_id":"qf-mod38-eulerthm-congruence","seed":202611076007,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":8,\"n\":11}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(11, 8, 10, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (11, 8, 10, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0008","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4afdf1cd23c0cab820627033678d619204df5058d5ba420e49e8c7e7a0d26829","family_id":"qf-mod38-eulerthm-congruence","seed":202611076008,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":11,\"n\":12}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(12, 11, 4, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (12, 11, 4, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0009","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6705fe0affde78b7222c9fdc094a172f0fdb72e717b31b1a15524c5bbd7d44e9","family_id":"qf-mod38-eulerthm-congruence","seed":202611076009,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":10,\"n\":13}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(13, 10, 12, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (13, 10, 12, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0010","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0643a54dd1d06e77e7df2aeadd89d9534262ca7e4521d8e8e6634311699cdd6d","family_id":"qf-mod38-eulerthm-congruence","seed":202611076010,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":11,\"n\":14}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(14, 11, 6, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (14, 11, 6, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0011","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a79e4bf42689053e0da517b1df0f2c47e1788356d4334f9de9ae9e0a36eb89a","family_id":"qf-mod38-eulerthm-congruence","seed":202611076011,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":13,\"n\":15}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(15, 13, 8, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (15, 13, 8, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0012","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"992bc6d95ad77230b9552217a8bf24fc82eeacbe609e542c1cc141cd786ca137","family_id":"qf-mod38-eulerthm-congruence","seed":202611076012,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":13,\"n\":16}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(16, 13, 8, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (16, 13, 8, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0013","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d8c94bb346cb3231c8e6ca4f48dd3cbd3cd1f1dd21fa98080bbaf4da4e8458b","family_id":"qf-mod38-eulerthm-congruence","seed":202611076013,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":14,\"n\":17}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(17, 14, 16, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (17, 14, 16, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0014","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"622473a4c109a0a056bcee0a888acdaf505ebdcf4c9382a156fc3668f16218f8","family_id":"qf-mod38-eulerthm-congruence","seed":202611076014,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":17,\"n\":18}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(18, 17, 6, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (18, 17, 6, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0015","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"679858394ab864f21f27ebb7cb0b9a4096fb08871fe39b5704c78e910e732543","family_id":"qf-mod38-eulerthm-congruence","seed":202611076015,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":16,\"n\":19}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(19, 16, 18, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (19, 16, 18, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0016","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58e7d09e3dbcf3f474a35bb0a77ca6cd6ee837de0e63f57fda3801e9eda1244e","family_id":"qf-mod38-eulerthm-congruence","seed":202611076016,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":17,\"n\":20}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(20, 17, 8, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (20, 17, 8, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0017","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea4bcf2b72df53f1c0196fe62bae9df407e02149a9d294c2f2d38323da99235a","family_id":"qf-mod38-eulerthm-congruence","seed":202611076017,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":19,\"n\":21}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(21, 19, 12, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (21, 19, 12, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0018","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"139823f16cb92aa301bd8ec896a3579605b53404054547f116822aeade00409f","family_id":"qf-mod38-eulerthm-congruence","seed":202611076018,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":19,\"n\":22}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(22, 19, 10, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (22, 19, 10, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0019","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cf5214de5a797f346972ee345f75a829bd0513eba90c3556b92633e56c074f8","family_id":"qf-mod38-eulerthm-congruence","seed":202611076019,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":20,\"n\":23}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(23, 20, 22, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (23, 20, 22, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0020","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53e757ffc21bb71b84ce165c11584d39e5a6e4f282f811ab8f8e36d9d23e54dd","family_id":"qf-mod38-eulerthm-congruence","seed":202611076020,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":24}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(24, 23, 8, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (24, 23, 8, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0021","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d44b86e5c5f6ec4687b1a763816c9d4224dd77e2ede52acf8ecf79dabbac5a7","family_id":"qf-mod38-eulerthm-congruence","seed":202611076021,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":22,\"n\":25}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(25, 22, 20, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (25, 22, 20, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0022","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c42477534741f16a7e77546618c67f071825ad0c35f1cdc365b2867f1c6b97b9","family_id":"qf-mod38-eulerthm-congruence","seed":202611076022,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":26}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(26, 23, 12, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (26, 23, 12, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0023","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da78d64d7c0a2253feb5902179b7497996d55db6d216f9b9dea7c5e49775e23f","family_id":"qf-mod38-eulerthm-congruence","seed":202611076023,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":25,\"n\":27}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(27, 25, 18, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (27, 25, 18, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0024","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"742392a3a56f4762e84f9b4146f69815381a7e7767136085cc83ef6ef4c508c1","family_id":"qf-mod38-eulerthm-congruence","seed":202611076024,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":25,\"n\":28}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(28, 25, 12, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (28, 25, 12, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0025","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"568586b8180bfc83c31faa4e4fdfa5a145dc25bde792d14906a6179081f1cdea","family_id":"qf-mod38-eulerthm-congruence","seed":202611076025,"semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":26,\"n\":29}"},"fields":{"title":"Euler Congruence Requires a Unit Residue, Not a Prime Modulus","stem":"Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).","answer":{"kind":"expression","value":"(29, 26, 28, 1, Yes)","explanation":"Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (29, 26, 28, 1, Yes)."},"skills":["Euler Congruence Requires a Unit Residue, Not a Prime Modulus"],"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-20261107-r06-euler-theorem-number-theory-0026","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a097c1495b417af1036624b02764596eac4163a853ee7663ea7adc4b0dbfd9e","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076026,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":2,\"n\":5}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4], [1, 2, 3, 4], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4], [1, 2, 3, 4], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0027","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e9b1c40b265ca4c6fa51a88ef5d05d1912cf95022ec96cc15dd13e90868429b","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076027,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":5,\"n\":6}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0028","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87132e2778a77f7469439d64e786c612d3de1718690736bad979e3e72f0aabaa","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076028,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":4,\"n\":7}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0029","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97924722c7ccc7ad8ed6815eba8e83cb64a0d76ab400f3a4e588859cd4514cc5","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076029,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":5,\"n\":8}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0030","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80327990574351da7be68cca334d287137e80cb8fb0762bbcdb354423d5f2a83","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076030,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":7,\"n\":9}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0031","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6550f3b5544a3e9d963ab58204bcf78a524958b2dcded3a3a473c5c73fba6ff5","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076031,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":7,\"n\":10}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0032","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2969b35a4d7e8265064a12d50f1d1979d23b8e69b08ddca578fe1c4c3ea76b0","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076032,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":8,\"n\":11}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0033","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c71a5559cfac72f2942ccced812f1040f97b112b56a6ae1f4bc9b7063ff0ce41","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076033,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":11,\"n\":12}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0034","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e68c235e59769a1a836286187b992b5b91615a2a2a28324be9a5cc9dbc213f1","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076034,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":10,\"n\":13}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0035","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdb709caaa75d39ce591d83519994e02ece0540d27de85d14df770d1fb129697","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076035,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":11,\"n\":14}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0036","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29e3518d9895a45265b831e006063be2ab7806b640d2524c0770a66a070d4ed3","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076036,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":13,\"n\":15}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0037","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e10c75fab3268fea75d78684633085eb0093e1d76b3ce10621e62eab25b65c59","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076037,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":13,\"n\":16}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0038","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40bffe11fd1e3ac7fd81121eb15a853e529a769d13802c82dabd12e5bb994fee","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076038,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":14,\"n\":17}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0039","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9725ddf396e85c02b31a8ea8b1e70d09291fc2765a45465f1608760a5c96d9e5","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076039,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":17,\"n\":18}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0040","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8783b0096e30d973c3286559eeb4705fe7e83f39042b71be9e1bc168d253cb0b","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076040,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":16,\"n\":19}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0041","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8528fe239bea5bbfd5604e3c147f0033e748537bc4dc85e5d5f6046278e1b16","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076041,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":17,\"n\":20}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0042","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f72d2eb2ca86b0c089b53f2d4e48d2f996bc36404e5ebe9992bf2c4b89ab6ac","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076042,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":19,\"n\":21}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0043","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b55f158459f92844dc26ca54589cc0a820fe02d31d592400dd91aff39d807d5","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076043,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":19,\"n\":22}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0044","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41deaacda375fd98fc4e17e87bd7f8edb2102223044ee596c152bd99629e1eb2","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076044,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":20,\"n\":23}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0045","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"653902dd07f617962da3ecaa0ca150f5d36244c5fd8a89745d5a8c0459271e21","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076045,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":23,\"n\":24}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0046","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"513515a3c9cee8702df0f4bcb666881db39297e3a6514613269411bfa20e7f4c","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076046,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":22,\"n\":25}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0047","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"530ae12faa0bef364bb90a617a9fc0c5e81e3d4cf2ad5198f2b1d0fc6e063694","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076047,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":23,\"n\":26}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0048","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b05b73d9cb21b462fd4b30f81a256f78675c5313883cf86e86e382bac4f3863a","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076048,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":25,\"n\":27}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0049","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f47487fab1994db168d1b523150fe1f7117e1d53d53f5937de43eae541be0066","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076049,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":25,\"n\":28}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0050","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8597de7a162277877a8de1a15604f3ece66a928137499b003ed42d9a0165072b","family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076050,"semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":26,\"n\":29}"},"fields":{"title":"Multiplication by a Unit Permutes the Complete Reduced Residues","stem":"Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues)","explanation":"List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues)."},"skills":["Multiplication by a Unit Permutes the Complete Reduced Residues"],"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-20261107-r06-euler-theorem-number-theory-0051","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3cba1bb9de18c6b9d5685022309b89b7c13be00c2c446e7ac657dbf1cc45176","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076051,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-2,\"n\":5}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=5≥2 and original integer base a=-2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0052","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75e82cfbe59f2c6c2a934ec0117c591479f9dcbbd448e41ebe5c5362c4b3e0a0","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076052,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-5,\"n\":6}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=6≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0053","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9c164db5687354a4d0f52ab2a5ca3d09aede29bb01962db37cdfa79fae860c8","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076053,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-4,\"n\":7}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=7≥2 and original integer base a=-4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0054","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b3a3cd8499a7ac2ebec9c653dded412a55273902314b15658f137ef8dd9eab0","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076054,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-5,\"n\":8}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=8≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0055","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14009ec5b757de7e26dc48d9e9b23c3888b4469f1cd0e815019d5a2cab8323f6","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076055,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-7,\"n\":9}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=9≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-7, 2, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 2, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0056","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07ad6d589305bbcea3352894dfdaf66a7cafd27dfbe5347f068c449cf22f85f6","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076056,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-7,\"n\":10}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=10≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0057","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46d514819b8a8b3fae65042a66e72e8d78934d80024b1c370a7cc0045ca7f2cb","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076057,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-8,\"n\":11}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=11≥2 and original integer base a=-8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0058","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff7a6ce4ee2f6b6e7ed61ba766b775a3743e9bcb7b5591f4a5402e8e13aff41e","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076058,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-11,\"n\":12}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=12≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0059","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88765fe8f9bd54267ad46bc9098a1409ff53910944d49b0618eccb8a39bdf463","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076059,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-10,\"n\":13}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=13≥2 and original integer base a=-10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0060","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b584793c0f049af00678dcf2aec28055eefd0a54c45d5d710f12dbc1fa84bd57","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076060,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-11,\"n\":14}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=14≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0061","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8341396f533e489562161bc172be88ba73a54b0757c01e0c806d43304123374f","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076061,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-13,\"n\":15}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=15≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0062","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28ca1c8ffdd01992d97fbed7359a910758077e92d67c61be72e79e60f6220370","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076062,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-13,\"n\":16}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=16≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0063","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54e5e69215a5ffa79516bca0de004e42594f927ecef0b66044fd0b1dce4b869a","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076063,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-14,\"n\":17}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=17≥2 and original integer base a=-14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0064","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c0ff9453f8fc8fa17291753a87f8cc2948343b663e18ea11a7d27df55cf212","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076064,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-17,\"n\":18}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=18≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0065","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d7eaaf46a202d9fcd2c6ce7f9d7e4a3e61de55134b4d650a98e119e110812d7","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076065,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-16,\"n\":19}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=19≥2 and original integer base a=-16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0066","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"199dfe450f3a818ec9b03a3351b3bb03d162bfb3ea43b6e181032fe9af87f6b5","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076066,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-17,\"n\":20}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=20≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0067","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ae68c593fe2573f8d22bec4046ae2d8ac9d213c041247b140fb745f8affe3d","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076067,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-19,\"n\":21}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=21≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0068","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"742139119e226695fef3373bee0e0ec54cebb03317aed7090ccada49979ffa68","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076068,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-19,\"n\":22}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=22≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0069","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad2573ea1439d22f5dba731bc0a5b1eeb86e0fece012ef0e3ae96773be2b41cd","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076069,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-20,\"n\":23}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=23≥2 and original integer base a=-20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0070","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b2806bf5cd3ba67a81ad0810ef6cd0148abbb0a7fee13d3e9e5ea94fe89ef7","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076070,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-23,\"n\":24}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=24≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0071","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95e4088e5debfcd1e637972e54475f7c0fd3212ccf384e12c575d1c796bfc096","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076071,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-22,\"n\":25}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=25≥2 and original integer base a=-22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0072","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aecc533cd9751cc1e00d5e7dbc17fcaf9a87684f37f517c587de3c72737779b","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076072,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-23,\"n\":26}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=26≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0073","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92f96d302676bde3bdcefa9e3bca6b0a3599452a74b2d5b763200058bb9b95a1","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076073,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-25,\"n\":27}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=27≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0074","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"610fffeb9f90bcc8340b27e29db929500bd55549045dfaafab85b5f71ce6801b","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076074,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-25,\"n\":28}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=28≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0075","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dc7b668c85d228c3b1244ea83b5cd567eacf3e604376e77a2cf9375b6eb56e9","family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076075,"semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-26,\"n\":29}"},"fields":{"title":"Normalize the Signed Base before Exact Modular Exponentiation","stem":"Use integer modulus n=29≥2 and original integer base a=-26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).","answer":{"kind":"expression","value":"(-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power)","explanation":"Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power)."},"skills":["Normalize the Signed Base before Exact Modular Exponentiation"],"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-20261107-r06-euler-theorem-number-theory-0076","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1cfc41580de2b9f342191d1b4b49a1fd7da63b7d7d60b997c497c89c5699d6","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076076,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":6,\"n\":6}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=6≥2 and original integer base a=6. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0077","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e72fa47e7a01ad8c7019bcfda51472f2b4eaa0d1ebbd2ea5eb2cfbbc754b66d6","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076077,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":7,\"n\":7}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=7≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0078","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85c40b04c100dad5df179064ee8126d9073a077929dcb1bd7d405244d5686409","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076078,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":8,\"n\":8}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=8≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0079","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e142c507806a8e70cf0225213232d218ef2e146fc99c75b878d3001bfe6fdb1","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076079,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":9,\"n\":9}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=9≥2 and original integer base a=9. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0080","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f0988727962c0449a169ee43fccc340f0248362adb8fb4aec20d67391041244","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076080,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":10,\"n\":10}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=10≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0081","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35d45fd3938bb9f3a62ad9bf654d69c2a42dcfdbd7375afc05af33f730f36110","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076081,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":11,\"n\":11}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=11≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0082","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b8fc39c422bf036136fc8022e71eea9ad3f8a05f3defb7fa13ee0d16f10a85","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076082,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":12,\"n\":12}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=12≥2 and original integer base a=12. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0083","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bf8877a2a290f7875a714e07dd25309273f06adeb0e2117ee55186398919fa3","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076083,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":13,\"n\":13}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=13≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0084","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7fc85085cdf77a437c6b52c04b0686c6e8f0ce34873aaaf334a9bb15ff538c2","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076084,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":14,\"n\":14}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=14≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0085","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dead634cde6fd18f238847782596d053aee3576c75d57644fbf064ad242d276b","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076085,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":15,\"n\":15}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=15≥2 and original integer base a=15. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0086","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3a4769a833cb4775de1fa21e8fcf8d84f792c0ddfdf7c0da2774145aafe0e77","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076086,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":16,\"n\":16}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=16≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0087","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff40d9294930e9ce41e146725eda89423350658699bce89ab54b30657399f4f","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076087,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":17,\"n\":17}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=17≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0088","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c91d1682293d4b2ea043a69b762d5f9137ca402be12babf271ce769ac5b00b0","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076088,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":18,\"n\":18}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=18≥2 and original integer base a=18. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0089","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f2f1f3a13ff7cb3a8dd09f313953814c8c17c0c295c4050f32703400c5fd50","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076089,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":19,\"n\":19}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=19≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0090","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67aec7ec76644b66722eebd760227c485892c00f223709b790ceeae94ec74a02","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076090,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":20,\"n\":20}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=20≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0091","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fe0d76fb7fe192b73783c991d4c4d32c3f9c1df52cc7c9fcbcbdbb0aa8ce387","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076091,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":21,\"n\":21}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=21≥2 and original integer base a=21. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0092","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0bebe09d7e5b2d7867941e4b57120146be533233290ada01644ed40beed0b23","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076092,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":22,\"n\":22}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=22≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0093","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a82f573bf8a2c2c7ac29c60151d75371c5d707199b9ea02b172ae30eedaa1461","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076093,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":23,\"n\":23}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=23≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0094","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"880d250660fad0a2a1b58f0c1f334066a16dc71517f4a215c729137424e6ae4b","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076094,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":24,\"n\":24}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=24≥2 and original integer base a=24. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0095","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"403198899ac8b64743a1efcb0d0df8043469dcd7c6d88bf9723a2477e94f152d","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076095,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":25,\"n\":25}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=25≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0096","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8563494a1863bd2320cf4985bf11785c4c1c3ceae39a1bacc7fef8b90df915eb","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076096,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":26,\"n\":26}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=26≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0097","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1feb7f085955a4104ad6471884e97586fd79276fc7884479d0ce89e2e6339d5a","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076097,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":27,\"n\":27}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=27≥2 and original integer base a=27. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0098","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc06ad6d867af721d4dd618f1758e4e8ce38c6acc9ff146c356c3f1f0dcd1be","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076098,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":28,\"n\":28}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=28≥2 and original integer base a=28. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0099","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb06dddd8331c29652a18283d3af8e18d3100b7f2a47501a5201a315bd49310a","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076099,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":29,\"n\":29}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=29≥2 and original integer base a=29. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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-20261107-r06-euler-theorem-number-theory-0100","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf7cc37f309faf9d459b8021ad486e3614abc1daa25373fe10a5532c1e6a410c","family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076100,"semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":30,\"n\":30}"},"fields":{"title":"A Nonunit Fails Euler Applicability, Not a Primality Test","stem":"Use integer modulus n=30≥2 and original integer base a=30. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).","answer":{"kind":"expression","value":"(30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation":"Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)."},"skills":["A Nonunit Fails Euler Applicability, Not a Primality Test"],"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."}}