{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261018-r03-gcd-coprime-0001","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0734f3bb9471f3980fab5636a219f97053f5f6b85e3540bff8ca5f9d82c6269","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183001,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":0,\"b\":0}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=0, b=0. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(0, 0, [], 0)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, [], 0)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0002","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96cb910a95c7ba89a193e9cb48aeb04670394d27897dd645e1df9dd53b877550","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183002,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":42,\"b\":17}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=42, b=17. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0003","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95b9fcf42bc92957d1ea8ef2c2095396a1992b1c434d56b391c7ff87b021e7f7","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183003,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":49,\"b\":22}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=49, b=22. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0004","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"976b069d4df41dcdb09d8afb15a41664081cbdfa54026514b68065b560dace39","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183004,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":56,\"b\":27}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=56, b=27. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0005","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7225aeedd656d45fae95a3fe8d5d0f4419e961e4a000510fb858f2e4470e8136","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183005,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":63,\"b\":32}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=63, b=32. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0006","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bdcc77b4d263fcf85fb48ac40f19b15d28cfc64e67d7fda16edb2f27ac71a10","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183006,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":70,\"b\":37}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=70, b=37. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0007","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61129c407021d5d887fb7dd55d59642f507d9579c4d53acbd1fe306fd85f33bf","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183007,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":77,\"b\":42}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=77, b=42. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0008","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d2b7edaf6d79ccdb6dfac5e2d575786c7d7d843e7147fa7e0ea6eece1ed850b","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183008,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":84,\"b\":47}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=84, b=47. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(84, 47, [[84, 47, 37], [47, 37, 10], [37, 10, 7], [10, 7, 3], [7, 3, 1], [3, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (84, 47, [[84, 47, 37], [47, 37, 10], [37, 10, 7], [10, 7, 3], [7, 3, 1], [3, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0009","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a730a097dd16b40f474eb23606e1e868a2b810ca38ffc8ba17242db74d5eaf97","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183009,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":91,\"b\":52}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=91, b=52. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0010","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5862b04de3c316d33e3f644cd2a57d58090fe15bd5355b6cff76d888f005bb6d","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183010,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":98,\"b\":57}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=98, b=57. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(98, 57, [[98, 57, 41], [57, 41, 16], [41, 16, 9], [16, 9, 7], [9, 7, 2], [7, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (98, 57, [[98, 57, 41], [57, 41, 16], [41, 16, 9], [16, 9, 7], [9, 7, 2], [7, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0011","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee82efdc2386e92e0061d46de4a7df5652a269e8859847873ab0648f62d79456","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183011,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":105,\"b\":62}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=105, b=62. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(105, 62, [[105, 62, 43], [62, 43, 19], [43, 19, 5], [19, 5, 4], [5, 4, 1], [4, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (105, 62, [[105, 62, 43], [62, 43, 19], [43, 19, 5], [19, 5, 4], [5, 4, 1], [4, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0012","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"955d774d44f7995737bd8cd5246bc94f0cc37ebf1c6a1af6c142d52466b33266","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183012,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":112,\"b\":67}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=112, b=67. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0013","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03c2067320059c5e40163021019a01b1e78923316ac0d3789ee95f9e3dbd642b","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183013,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":119,\"b\":72}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=119, b=72. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0014","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93d4c3a492d0ec1d10d91a949d68528080aa2aa81ccc124fe9392e2854540df4","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183014,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":126,\"b\":77}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=126, b=77. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0015","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9d18d1c200a6c9c0a42e23dc95dcfafccb08636e8c38e756cbea7c3ebe09aff","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183015,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":133,\"b\":82}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=133, b=82. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(133, 82, [[133, 82, 51], [82, 51, 31], [51, 31, 20], [31, 20, 11], [20, 11, 9], [11, 9, 2], [9, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (133, 82, [[133, 82, 51], [82, 51, 31], [51, 31, 20], [31, 20, 11], [20, 11, 9], [11, 9, 2], [9, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0016","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fafd4a4d2c9691c6306fa5dd91181cab1c58223c5f6d93fb8949ef797f7167d7","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183016,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":140,\"b\":87}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=140, b=87. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(140, 87, [[140, 87, 53], [87, 53, 34], [53, 34, 19], [34, 19, 15], [19, 15, 4], [15, 4, 3], [4, 3, 1], [3, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (140, 87, [[140, 87, 53], [87, 53, 34], [53, 34, 19], [34, 19, 15], [19, 15, 4], [15, 4, 3], [4, 3, 1], [3, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0017","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc48bcf1fdb9f57df413ac2918cbaafe1d0427150291f7dd73afbb660fb5c254","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183017,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":147,\"b\":92}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=147, b=92. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0018","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97e3c560ec66c8f59d8418e69203d0695bc0e8e2300ef59dd1f20601cfe96e6a","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183018,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":154,\"b\":97}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=154, b=97. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(154, 97, [[154, 97, 57], [97, 57, 40], [57, 40, 17], [40, 17, 6], [17, 6, 5], [6, 5, 1], [5, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (154, 97, [[154, 97, 57], [97, 57, 40], [57, 40, 17], [40, 17, 6], [17, 6, 5], [6, 5, 1], [5, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0019","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"545b2cceab3ded845068912890776ca95bd5ae2049236343d2e3746e30f27009","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183019,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":161,\"b\":102}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=161, b=102. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0020","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"154ed8af9a1656500a68320b46e6b85826e478825acbadbef6f84cbf3c9cdade","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183020,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":168,\"b\":107}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=168, b=107. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0021","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea50ddc3039df2a7bcba2850f10d04ce0e95a43000e041171285148d8b7ac8b","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183021,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":175,\"b\":112}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=175, b=112. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0022","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba7da027973af10bcddbef20002529dd462a297a62fdf0608f3b6935a4c1006","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183022,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":182,\"b\":117}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=182, b=117. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0023","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dd92fe09916f63c6c473188211383747c3453ae4fcd25f6367b13441526c80a","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183023,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":189,\"b\":122}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=189, b=122. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(189, 122, [[189, 122, 67], [122, 67, 55], [67, 55, 12], [55, 12, 7], [12, 7, 5], [7, 5, 2], [5, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (189, 122, [[189, 122, 67], [122, 67, 55], [67, 55, 12], [55, 12, 7], [12, 7, 5], [7, 5, 2], [5, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0024","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091d19bc7738dcfe2a5bb5987b6a6f6049a69d0995d5d63274a67049d4cf5645","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183024,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":196,\"b\":127}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=196, b=127. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(196, 127, [[196, 127, 69], [127, 69, 58], [69, 58, 11], [58, 11, 3], [11, 3, 2], [3, 2, 1], [2, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (196, 127, [[196, 127, 69], [127, 69, 58], [69, 58, 11], [58, 11, 3], [11, 3, 2], [3, 2, 1], [2, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0025","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8226ba969c53f7882769d09c84e58f0021c7dd2a59f7866bc22f4ab0e74f4e8a","family_id":"qf-ext18-gcd-euclid-trace","seed":202610183025,"semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":203,\"b\":132}"},"fields":{"title":"Exact Euclidean Remainder Trace and Natural GCD","stem":"Use natural gcd with a=203, b=132. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.","answer":{"kind":"expression","value":"(203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1)","explanation":"Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1)."},"skills":["Exact Euclidean Remainder Trace and Natural GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0026","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01770de2c22e41f74410d5d620c41f16e86adf1396de3483b6f5117b9f450536","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183026,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":36,\"b\":24,\"d\":12}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=36, b=24, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(36, 24, 12, Yes, Yes, Yes, 12, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 24, 12, Yes, Yes, Yes, 12, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0027","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba2629c0d3acded311e16002bd99f18d981d5c932fbe6fb90bbda438797edeb0","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183027,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":40,\"b\":30,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=40, b=30, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(40, 30, 1, Yes, Yes, No, 10, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 30, 1, Yes, Yes, No, 10, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0028","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d04a73fa35af1767cce7dacae52355733c383808ddd289bf4fe1196bae9ba0ef","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183028,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":44,\"b\":36,\"d\":5}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=44, b=36, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(44, 36, 5, No, No, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (44, 36, 5, No, No, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0029","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dc632e60b49a46ceb0130720b11b3bd2d65b5b8994c82f5556348ae909103c2","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183029,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":48,\"b\":42,\"d\":6}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=48, b=42, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(48, 42, 6, Yes, Yes, Yes, 6, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (48, 42, 6, Yes, Yes, Yes, 6, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0030","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ccbf2303106b78c6e5424f996e581593684ff87ebc7a0c76101b04abe642924","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183030,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":52,\"b\":48,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=52, b=48, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(52, 48, 1, Yes, Yes, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (52, 48, 1, Yes, Yes, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0031","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1ff82c6da58b081d25fa26bc2092921bb98ab64a5e39591bd07e3e76142ad3","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183031,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":56,\"b\":54,\"d\":3}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=56, b=54, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(56, 54, 3, No, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (56, 54, 3, No, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0032","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"152836be46144b050b40c618391130928a36978ead78bbb86408e97c89c205c6","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183032,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":60,\"b\":60,\"d\":60}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=60, b=60, candidate d=60; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(60, 60, 60, Yes, Yes, Yes, 60, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (60, 60, 60, Yes, Yes, Yes, 60, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0033","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0566edf9280f9cc1ce5b7831c6b7efaf92be04e86c4ce3c317db57c9f383c8de","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183033,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":64,\"b\":66,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=64, b=66, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(64, 66, 1, Yes, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (64, 66, 1, Yes, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0034","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef26c0196ec2376c8a59ec0aaaf7a2d3be09cec0aa0bca761554b7e4d3dffc4f","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183034,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":68,\"b\":72,\"d\":5}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=68, b=72, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(68, 72, 5, No, No, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (68, 72, 5, No, No, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0035","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"888ab4fe0712514edc0cd8fa4344df127a7094d6fc9a65bad26777c7abae7a5b","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183035,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":72,\"b\":78,\"d\":6}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=72, b=78, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(72, 78, 6, Yes, Yes, Yes, 6, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (72, 78, 6, Yes, Yes, Yes, 6, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0036","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d39dcc4c33bbcfe6aaff7debd290baf03967594a9e22921854c2184e39fcc4b","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183036,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":76,\"b\":84,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=76, b=84, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(76, 84, 1, Yes, Yes, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (76, 84, 1, Yes, Yes, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0037","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad4789ac97bf49841bf8b5ac96aa8c93a22cd1e6cc252e9ecb1a79f8f213c94c","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183037,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":80,\"b\":90,\"d\":11}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=80, b=90, candidate d=11; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(80, 90, 11, No, No, No, 10, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (80, 90, 11, No, No, No, 10, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0038","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9387ba3d08be214c70a9122e15502f6ccbaf29fc4d00bd041822dbd37e70c9e7","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183038,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":84,\"b\":96,\"d\":12}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=84, b=96, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(84, 96, 12, Yes, Yes, Yes, 12, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (84, 96, 12, Yes, Yes, Yes, 12, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0039","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c1c51e31b9b7cac3d8dd4ea9ef63c501b28fc99f01c1237753f6e7f1ba9b81c","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183039,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":88,\"b\":102,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=88, b=102, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(88, 102, 1, Yes, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (88, 102, 1, Yes, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0040","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4698c0e0a83ee5749c2b5a29e5de03c1d2260dcada56323e5c9637bac5bbdc67","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183040,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":92,\"b\":108,\"d\":5}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=92, b=108, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(92, 108, 5, No, No, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (92, 108, 5, No, No, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0041","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c1283b97f6ad16d88bce674df61861ca95be3624b6b340fe3646f8953ce7fe8","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183041,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":96,\"b\":114,\"d\":6}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=96, b=114, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(96, 114, 6, Yes, Yes, Yes, 6, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (96, 114, 6, Yes, Yes, Yes, 6, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0042","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64c7f121d161be982f76650b4a798f2741d0fade28b5115739e9c2f38b57ef4d","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183042,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":100,\"b\":120,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=100, b=120, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(100, 120, 1, Yes, Yes, No, 20, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (100, 120, 1, Yes, Yes, No, 20, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0043","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e98063dcb25a14682f162404500ed3683c7011c96014859116e10fee7fe7f252","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183043,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":104,\"b\":126,\"d\":3}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=104, b=126, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(104, 126, 3, No, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (104, 126, 3, No, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0044","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88938ef9f39de3fdd6a02815d73726a481fad9a7028c0a0189738df1c3fea98e","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183044,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":108,\"b\":132,\"d\":12}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=108, b=132, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(108, 132, 12, Yes, Yes, Yes, 12, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (108, 132, 12, Yes, Yes, Yes, 12, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0045","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9e406b59e7f48de8e41f00d91f7dc9bcbf9c9d70de36fc5f0fdd320be01e244","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183045,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":112,\"b\":138,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=112, b=138, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(112, 138, 1, Yes, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (112, 138, 1, Yes, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0046","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2e3b75de8e7073ebcce80b9aefb5b2b7e4e46e6555e6afdac2f11151b5b70da","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183046,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":116,\"b\":144,\"d\":5}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=116, b=144, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(116, 144, 5, No, No, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (116, 144, 5, No, No, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0047","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98466d9070f62eb48c1a4401fb5b86c21d0fd441b7fa53bdf9373c458550388","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183047,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":120,\"b\":150,\"d\":30}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=120, b=150, candidate d=30; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(120, 150, 30, Yes, Yes, Yes, 30, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (120, 150, 30, Yes, Yes, Yes, 30, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0048","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc74938f71d59eec8c994906e9a6b1acb20f7116c1ca761eb2d85a0a6e68a23c","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183048,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":124,\"b\":156,\"d\":1}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=124, b=156, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(124, 156, 1, Yes, Yes, No, 4, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (124, 156, 1, Yes, Yes, No, 4, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0049","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1159a17f4845acebfdcd2899388af427f4273db52349b7a55d15acd6140eccc","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183049,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":128,\"b\":162,\"d\":3}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=128, b=162, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(128, 162, 3, No, Yes, No, 2, No)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (128, 162, 3, No, Yes, No, 2, No)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0050","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1999b11da8d836bbe8632f34206ea702632d2b273b1eaedfd31cd124f47de60","family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183050,"semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":132,\"b\":168,\"d\":12}"},"fields":{"title":"A Common Divisor versus the Actual GCD","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=132, b=168, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).","answer":{"kind":"expression","value":"(132, 168, 12, Yes, Yes, Yes, 12, Yes)","explanation":"Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (132, 168, 12, Yes, Yes, Yes, 12, Yes)."},"skills":["A Common Divisor versus the Actual GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0051","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0b42ccb71023c2c925392493354509df3e1ff576d5f158d26922591a9e9c5a7","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183051,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":0,\"b\":1}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=0, b=1 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(0, 1, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0052","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15df5bfea93aa595d200ed757456190d5ecf52f74c7e83643d8971212762a01","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183052,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":1,\"b\":3}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=1, b=3 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(1, 3, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0053","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"059f072e358dfd05ba2f1998625109b82e07fffe5a2885ba11cac36375e052dc","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183053,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":2,\"b\":5}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=2, b=5 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(2, 5, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0054","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8e5627becbdaaae440373ce4f91a7530ae1002df27284743cafb47cd9d45212","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183054,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":3,\"b\":7}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=3, b=7 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(3, 7, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 7, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0055","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48d544f22fed168794feacd990bd849a891cb7de47277600bd1d809762a94ea","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183055,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":4,\"b\":9}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=4, b=9 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(4, 9, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 9, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0056","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c32602c19cc9411fcb23c0f134978f32ac116f4e86343cc65debd000d23af40","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183056,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":5,\"b\":11}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=5, b=11 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(5, 11, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 11, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0057","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14354b748e98e855b5a15aeeb98fe820fca84122b72a2a4688337e6766f5f803","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183057,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":6,\"b\":13}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=6, b=13 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(6, 13, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 13, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0058","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7c86c3677339053d06ffd6905d88ecdb454567486a6c008511db9145c21f9c4","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183058,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":7,\"b\":15}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=7, b=15 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(7, 15, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 15, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0059","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5907c8226b35ec9fc60cb8d03c27a4debfdd766c64fa0e1f39903b7da489c79","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183059,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":8,\"b\":17}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=8, b=17 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(8, 17, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 17, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0060","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5196204b028cecfea55aa2e7d0c453141af63892ed25863b318657037ba5368e","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183060,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":9,\"b\":19}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=9, b=19 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(9, 19, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 19, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0061","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"601ff75007acab5bd0fc50f1908369b7bc806940c86fb41b89c5e52d5a769657","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183061,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":10,\"b\":21}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=10, b=21 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(10, 21, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 21, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0062","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f145bc4e0abe717574aff86a501a2abd80c56e81d1daaafe808fba37d5fa88","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183062,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":11,\"b\":23}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=11, b=23 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(11, 23, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 23, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0063","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3315e563ea2ad5c04a7c55ff18e6408c2d3327e6521e43cf50405ab15e270cce","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183063,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":12,\"b\":25}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=12, b=25 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(12, 25, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 25, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0064","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a0d31ce923a777e4e754c68bb10c8dfd4b58c5a5d138508051e73cb8b32bae1","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183064,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":13,\"b\":27}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=13, b=27 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(13, 27, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 27, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0065","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd92bf8182672224c5effb72b9ae53342c7b95615d39663f9f34d12aca4ec756","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183065,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":14,\"b\":29}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=14, b=29 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(14, 29, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 29, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0066","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39e09c0411111387c94c5c957d20245a239e82fc73e183dee5f72e5618a4322","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183066,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":15,\"b\":31}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=15, b=31 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(15, 31, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 31, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0067","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8779f7edd9897ff6043e45b1863671f86a8d34c92d7d530b5a1af0b3c68ead6e","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183067,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":16,\"b\":33}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=16, b=33 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(16, 33, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 33, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0068","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fbccd6b75ae5b7af5f5f872e1acf4f8152ddb2ac3cfffac21384aa6314f112f","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183068,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":17,\"b\":35}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=17, b=35 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(17, 35, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 35, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0069","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1253518f9880bcd764dc7590416599c1a0e12f32650b2befa160d64d7d50849d","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183069,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":18,\"b\":37}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=18, b=37 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(18, 37, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 37, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0070","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6377928c88f6ed45b3d85745c1947813fce9232976f542dea1763d79d081196","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183070,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":19,\"b\":39}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=19, b=39 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(19, 39, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 39, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0071","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09b54a938070f6c16e43d8d78d6039d91d13cde452f7e6c5bc2da50af563b9e7","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183071,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":20,\"b\":41}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=20, b=41 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(20, 41, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 41, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0072","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1cd05799a70af77b176fbd20fcad6e1c665d3703beae6913a45f1b2a041a2c3","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183072,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":21,\"b\":43}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=21, b=43 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(21, 43, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 43, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0073","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee1ef21835c588472f8410e7385ac6d36c47b37bcff93bffbad2b028b186806","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183073,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":22,\"b\":45}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=22, b=45 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(22, 45, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 45, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0074","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85f9c0384703d12927b6a024b601b08a72c1b0a0562757cddfafc6aab53a20f4","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183074,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":23,\"b\":47}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=23, b=47 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(23, 47, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 47, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0075","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9767cdbc2f964a64d512ac7310951e7fed89d3ba98cad79b10a11616921436c8","family_id":"qf-ext18-gcd-coprime-classification","seed":202610183075,"semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":24,\"b\":49}"},"fields":{"title":"Natural GCD and Coprimality Including Zero","stem":"a=24, b=49 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.","answer":{"kind":"expression","value":"(24, 49, 1, Yes)","explanation":"Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 49, 1, Yes)."},"skills":["Natural GCD and Coprimality Including Zero"],"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-20261013-advance-20261018-r03-gcd-coprime-0076","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"664b5d32a460f7d72eebe500e49f4336bda3a99e86c2110b6e2cafdaa1f7a016","family_id":"qf-ext18-gcd-associative-triple","seed":202610183076,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":12,\"b\":18,\"c\":30}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=12, b=18, c=30, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(12, 18, 30, 6, 6, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 18, 30, 6, 6, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0077","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcc8ea621d09af7c01d62f45da87ad2bdd6fae614f4bc450d21e532222d0899d","family_id":"qf-ext18-gcd-associative-triple","seed":202610183077,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":13,\"b\":20,\"c\":33}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=13, b=20, c=33, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(13, 20, 33, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 20, 33, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0078","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e48a8e06410b4bef3c80df485f3b6f802886efd5eea6e0b6756b9afe78bfd88","family_id":"qf-ext18-gcd-associative-triple","seed":202610183078,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":14,\"b\":22,\"c\":36}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=14, b=22, c=36, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(14, 22, 36, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 22, 36, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0079","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8e2cc0f888622a95fdb10e73211f46a855ea6527ecebb5fca2e4bb1964923f3","family_id":"qf-ext18-gcd-associative-triple","seed":202610183079,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":15,\"b\":24,\"c\":39}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=15, b=24, c=39, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(15, 24, 39, 3, 3, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 24, 39, 3, 3, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0080","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af0a91e32b87f7e2d30e10604829ac3b2a0081eb0bf343c49cc3417c72ab2f73","family_id":"qf-ext18-gcd-associative-triple","seed":202610183080,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":16,\"b\":26,\"c\":42}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=16, b=26, c=42, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(16, 26, 42, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 26, 42, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0081","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2757cbc434dd4c5a68099e9f493f3123d6ec5aa8af557bba8bff3ec77835c7b","family_id":"qf-ext18-gcd-associative-triple","seed":202610183081,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":17,\"b\":28,\"c\":45}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=17, b=28, c=45, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(17, 28, 45, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 28, 45, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0082","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d598142a8e86a932f1201d107903d4c62f6f1f8054e8568b144f1a02ead6c30","family_id":"qf-ext18-gcd-associative-triple","seed":202610183082,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":18,\"b\":30,\"c\":48}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=18, b=30, c=48, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(18, 30, 48, 6, 6, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 30, 48, 6, 6, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0083","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b935dcba35fc4cea42472521b81568db6c5e4653a38dbbd543b1d269bef37fb2","family_id":"qf-ext18-gcd-associative-triple","seed":202610183083,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":19,\"b\":32,\"c\":51}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=19, b=32, c=51, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(19, 32, 51, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 32, 51, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0084","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd59b1043b81e8142a786f0114ec58e4b96c85b89ad3ba91c70857d06fd4f2f0","family_id":"qf-ext18-gcd-associative-triple","seed":202610183084,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":20,\"b\":34,\"c\":54}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=20, b=34, c=54, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(20, 34, 54, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 34, 54, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0085","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973938429101914f4e0fd609e9965877ecdddb238afd7c6a069c2d84f3cbc87d","family_id":"qf-ext18-gcd-associative-triple","seed":202610183085,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":21,\"b\":36,\"c\":57}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=21, b=36, c=57, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(21, 36, 57, 3, 3, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 36, 57, 3, 3, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0086","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92576e315c425e67bbbcaad5f0abc870dd6eb7915da187d0a6d5235a44d24b69","family_id":"qf-ext18-gcd-associative-triple","seed":202610183086,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":22,\"b\":38,\"c\":60}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=22, b=38, c=60, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(22, 38, 60, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 38, 60, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0087","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56f6b569c99861a2cff3dfeb43310a7692affd35346060c6ae6f2455217bfe7a","family_id":"qf-ext18-gcd-associative-triple","seed":202610183087,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":23,\"b\":40,\"c\":63}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=23, b=40, c=63, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(23, 40, 63, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 40, 63, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0088","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff1d7c7287e6a1fd5e8385a100078315f6a205a6b838a7b7e787553bcdc3ea93","family_id":"qf-ext18-gcd-associative-triple","seed":202610183088,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":24,\"b\":42,\"c\":66}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=24, b=42, c=66, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(24, 42, 66, 6, 6, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 42, 66, 6, 6, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0089","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b34d71b515c5c455c4382eba6438932989ca5e1617a1836e2fb4147a40631117","family_id":"qf-ext18-gcd-associative-triple","seed":202610183089,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":25,\"b\":44,\"c\":69}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=25, b=44, c=69, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(25, 44, 69, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, 44, 69, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0090","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3848aa561cf67a29fb6aaa53bec0f98713127132d696403e49f41fa3101468b1","family_id":"qf-ext18-gcd-associative-triple","seed":202610183090,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":26,\"b\":46,\"c\":72}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=26, b=46, c=72, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(26, 46, 72, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (26, 46, 72, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0091","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4edb64f5c7a7760679485a64d5cde1d85b6c37324c3a2c3f73c742c167d73616","family_id":"qf-ext18-gcd-associative-triple","seed":202610183091,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":27,\"b\":48,\"c\":75}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=27, b=48, c=75, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(27, 48, 75, 3, 3, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (27, 48, 75, 3, 3, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0092","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad0a508984c7c0dfd70f2757b5cf8e9df5d47f954ab110469bda022a5a3d0fe4","family_id":"qf-ext18-gcd-associative-triple","seed":202610183092,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":28,\"b\":50,\"c\":78}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=28, b=50, c=78, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(28, 50, 78, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (28, 50, 78, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0093","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fa31a2769ca89ff598d8d11b3e9e156c2db7d9b49e2e9d61c44fc0d9fd6d882","family_id":"qf-ext18-gcd-associative-triple","seed":202610183093,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":29,\"b\":52,\"c\":81}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=29, b=52, c=81, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(29, 52, 81, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (29, 52, 81, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0094","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ccd6df1131d055dc0a5217c785cec027cba06eefdf5e3ecef0d15252bdbd56","family_id":"qf-ext18-gcd-associative-triple","seed":202610183094,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":30,\"b\":54,\"c\":84}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=30, b=54, c=84, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(30, 54, 84, 6, 6, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, 54, 84, 6, 6, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0095","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6108eb844a8216d6b50b3aea9690e3d1ae3cbc6443c7ea2ee8b98afbf9cf85b","family_id":"qf-ext18-gcd-associative-triple","seed":202610183095,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":31,\"b\":56,\"c\":87}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=31, b=56, c=87, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(31, 56, 87, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (31, 56, 87, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0096","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b68b2c81ec8e4e5e137ca28b9737467299db6fad1471bf9b6cd4ccaa5757a23","family_id":"qf-ext18-gcd-associative-triple","seed":202610183096,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":32,\"b\":58,\"c\":90}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=32, b=58, c=90, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(32, 58, 90, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (32, 58, 90, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0097","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"523d4cc6420da520365c987b38a2e316a2713782b333330d037200d718a530c2","family_id":"qf-ext18-gcd-associative-triple","seed":202610183097,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":33,\"b\":60,\"c\":93}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=33, b=60, c=93, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(33, 60, 93, 3, 3, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, 60, 93, 3, 3, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0098","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25360cd2732af6b91d2aa1ae9c45b9c6df4c8f8907b491329578878846cf0bf9","family_id":"qf-ext18-gcd-associative-triple","seed":202610183098,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":34,\"b\":62,\"c\":96}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=34, b=62, c=96, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(34, 62, 96, 2, 2, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, 62, 96, 2, 2, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0099","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2545c42ff7a40c476d00113027aec6f111f714f38c714d92a8bf0ecdc5226d8","family_id":"qf-ext18-gcd-associative-triple","seed":202610183099,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":35,\"b\":64,\"c\":99}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=35, b=64, c=99, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(35, 64, 99, 1, 1, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (35, 64, 99, 1, 1, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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-20261013-advance-20261018-r03-gcd-coprime-0100","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c369d721eaea8866e4d6df04c72914278ad8af1f5642340a3bd281f8f5cac8d","family_id":"qf-ext18-gcd-associative-triple","seed":202610183100,"semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":36,\"b\":66,\"c\":102}"},"fields":{"title":"Both Parenthesizations of a Three-Input GCD","stem":"Natural inputs a=36, b=66, c=102, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).","answer":{"kind":"expression","value":"(36, 66, 102, 6, 6, Yes)","explanation":"Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 66, 102, 6, 6, Yes)."},"skills":["Both Parenthesizations of a Three-Input GCD"],"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."}}