{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0001","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":0,\"b\":0}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, [], 0)","explanation_zh":"擴展基礎核對=(0, 0, [], 0)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183001,"parameters":{"a":0,"b":0,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":0,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0002","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":42,\"b\":17}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183002,"parameters":{"a":42,"b":17,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":42,\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0003","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":49,\"b\":22}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183003,"parameters":{"a":49,"b":22,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":49,\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0004","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":56,\"b\":27}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183004,"parameters":{"a":56,"b":27,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":56,\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0005","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":63,\"b\":32}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183005,"parameters":{"a":63,"b":32,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":63,\"b\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0006","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":70,\"b\":37}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183006,"parameters":{"a":70,"b":37,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":70,\"b\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0007","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":77,\"b\":42}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7)","explanation_zh":"擴展基礎核對=(77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183007,"parameters":{"a":77,"b":42,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":77,\"b\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0008","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":84,\"b\":47}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(84, 47, [[84, 47, 37], [47, 37, 10], [37, 10, 7], [10, 7, 3], [7, 3, 1], [3, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183008,"parameters":{"a":84,"b":47,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":84,\"b\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0009","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":91,\"b\":52}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13)","explanation_zh":"擴展基礎核對=(91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183009,"parameters":{"a":91,"b":52,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":91,\"b\":52}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0010","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":98,\"b\":57}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(98, 57, [[98, 57, 41], [57, 41, 16], [41, 16, 9], [16, 9, 7], [9, 7, 2], [7, 2, 1], [2, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183010,"parameters":{"a":98,"b":57,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":98,\"b\":57}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0011","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":105,\"b\":62}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(105, 62, [[105, 62, 43], [62, 43, 19], [43, 19, 5], [19, 5, 4], [5, 4, 1], [4, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183011,"parameters":{"a":105,"b":62,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":105,\"b\":62}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0012","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":112,\"b\":67}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183012,"parameters":{"a":112,"b":67,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":112,\"b\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0013","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":119,\"b\":72}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183013,"parameters":{"a":119,"b":72,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":119,\"b\":72}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0014","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":126,\"b\":77}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7)","explanation_zh":"擴展基礎核對=(126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183014,"parameters":{"a":126,"b":77,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":126,\"b\":77}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0015","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":133,\"b\":82}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(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)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183015,"parameters":{"a":133,"b":82,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":133,\"b\":82}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0016","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":140,\"b\":87}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(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)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183016,"parameters":{"a":140,"b":87,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":140,\"b\":87}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0017","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":147,\"b\":92}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183017,"parameters":{"a":147,"b":92,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":147,\"b\":92}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0018","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":154,\"b\":97}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(154, 97, [[154, 97, 57], [97, 57, 40], [57, 40, 17], [40, 17, 6], [17, 6, 5], [6, 5, 1], [5, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183018,"parameters":{"a":154,"b":97,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":154,\"b\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0019","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":161,\"b\":102}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183019,"parameters":{"a":161,"b":102,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":161,\"b\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0020","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":168,\"b\":107}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183020,"parameters":{"a":168,"b":107,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":168,\"b\":107}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0021","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":175,\"b\":112}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7)","explanation_zh":"擴展基礎核對=(175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183021,"parameters":{"a":175,"b":112,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":175,\"b\":112}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0022","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":182,\"b\":117}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13)","explanation_zh":"擴展基礎核對=(182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183022,"parameters":{"a":182,"b":117,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":182,\"b\":117}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0023","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":189,\"b\":122}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(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)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183023,"parameters":{"a":189,"b":122,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":189,\"b\":122}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0024","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":196,\"b\":127}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","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_zh":"擴展基礎核對=(196, 127, [[196, 127, 69], [127, 69, 58], [69, 58, 11], [58, 11, 3], [11, 3, 2], [3, 2, 1], [2, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183024,"parameters":{"a":196,"b":127,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":196,\"b\":127}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0025","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-euclid-trace；資料={\"a\":203,\"b\":132}。請依 [a,b,euclid_rows,gcd] 的欄序回答。","answer":{"kind":"expression","value":"(203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1)","explanation_zh":"擴展基礎核對=(203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-euclid-trace","seed":202610183025,"parameters":{"a":203,"b":132,"domain":"Nat_gcd","operation":"gcd-euclid-trace","semantic_template":"qf-ext18-gcd-euclid-trace/{\"a\":203,\"b\":132}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0026","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":36,\"b\":24,\"d\":12}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(36, 24, 12, 是, 是, 是, 12, 是)","explanation_zh":"擴展基礎核對=(36, 24, 12, 是, 是, 是, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183026,"parameters":{"a":36,"b":24,"d":12,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":36,\"b\":24,\"d\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0027","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":40,\"b\":30,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(40, 30, 1, 是, 是, 否, 10, 否)","explanation_zh":"擴展基礎核對=(40, 30, 1, 是, 是, 否, 10, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183027,"parameters":{"a":40,"b":30,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":40,\"b\":30,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0028","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":44,\"b\":36,\"d\":5}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(44, 36, 5, 否, 否, 否, 4, 否)","explanation_zh":"擴展基礎核對=(44, 36, 5, 否, 否, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183028,"parameters":{"a":44,"b":36,"d":5,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":44,\"b\":36,\"d\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0029","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":48,\"b\":42,\"d\":6}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(48, 42, 6, 是, 是, 是, 6, 是)","explanation_zh":"擴展基礎核對=(48, 42, 6, 是, 是, 是, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183029,"parameters":{"a":48,"b":42,"d":6,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":48,\"b\":42,\"d\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0030","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":52,\"b\":48,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(52, 48, 1, 是, 是, 否, 4, 否)","explanation_zh":"擴展基礎核對=(52, 48, 1, 是, 是, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183030,"parameters":{"a":52,"b":48,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":52,\"b\":48,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0031","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":56,\"b\":54,\"d\":3}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(56, 54, 3, 否, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(56, 54, 3, 否, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183031,"parameters":{"a":56,"b":54,"d":3,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":56,\"b\":54,\"d\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0032","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":60,\"b\":60,\"d\":60}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(60, 60, 60, 是, 是, 是, 60, 是)","explanation_zh":"擴展基礎核對=(60, 60, 60, 是, 是, 是, 60, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183032,"parameters":{"a":60,"b":60,"d":60,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":60,\"b\":60,\"d\":60}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0033","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":64,\"b\":66,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(64, 66, 1, 是, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(64, 66, 1, 是, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183033,"parameters":{"a":64,"b":66,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":64,\"b\":66,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0034","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":68,\"b\":72,\"d\":5}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(68, 72, 5, 否, 否, 否, 4, 否)","explanation_zh":"擴展基礎核對=(68, 72, 5, 否, 否, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183034,"parameters":{"a":68,"b":72,"d":5,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":68,\"b\":72,\"d\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0035","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":72,\"b\":78,\"d\":6}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(72, 78, 6, 是, 是, 是, 6, 是)","explanation_zh":"擴展基礎核對=(72, 78, 6, 是, 是, 是, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183035,"parameters":{"a":72,"b":78,"d":6,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":72,\"b\":78,\"d\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0036","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":76,\"b\":84,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(76, 84, 1, 是, 是, 否, 4, 否)","explanation_zh":"擴展基礎核對=(76, 84, 1, 是, 是, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183036,"parameters":{"a":76,"b":84,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":76,\"b\":84,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0037","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":80,\"b\":90,\"d\":11}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(80, 90, 11, 否, 否, 否, 10, 否)","explanation_zh":"擴展基礎核對=(80, 90, 11, 否, 否, 否, 10, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183037,"parameters":{"a":80,"b":90,"d":11,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":80,\"b\":90,\"d\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0038","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":84,\"b\":96,\"d\":12}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(84, 96, 12, 是, 是, 是, 12, 是)","explanation_zh":"擴展基礎核對=(84, 96, 12, 是, 是, 是, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183038,"parameters":{"a":84,"b":96,"d":12,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":84,\"b\":96,\"d\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0039","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":88,\"b\":102,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(88, 102, 1, 是, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(88, 102, 1, 是, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183039,"parameters":{"a":88,"b":102,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":88,\"b\":102,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0040","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":92,\"b\":108,\"d\":5}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(92, 108, 5, 否, 否, 否, 4, 否)","explanation_zh":"擴展基礎核對=(92, 108, 5, 否, 否, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183040,"parameters":{"a":92,"b":108,"d":5,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":92,\"b\":108,\"d\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0041","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":96,\"b\":114,\"d\":6}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(96, 114, 6, 是, 是, 是, 6, 是)","explanation_zh":"擴展基礎核對=(96, 114, 6, 是, 是, 是, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183041,"parameters":{"a":96,"b":114,"d":6,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":96,\"b\":114,\"d\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0042","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":100,\"b\":120,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(100, 120, 1, 是, 是, 否, 20, 否)","explanation_zh":"擴展基礎核對=(100, 120, 1, 是, 是, 否, 20, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183042,"parameters":{"a":100,"b":120,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":100,\"b\":120,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0043","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":104,\"b\":126,\"d\":3}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(104, 126, 3, 否, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(104, 126, 3, 否, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183043,"parameters":{"a":104,"b":126,"d":3,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":104,\"b\":126,\"d\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0044","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":108,\"b\":132,\"d\":12}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(108, 132, 12, 是, 是, 是, 12, 是)","explanation_zh":"擴展基礎核對=(108, 132, 12, 是, 是, 是, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183044,"parameters":{"a":108,"b":132,"d":12,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":108,\"b\":132,\"d\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0045","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":112,\"b\":138,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(112, 138, 1, 是, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(112, 138, 1, 是, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183045,"parameters":{"a":112,"b":138,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":112,\"b\":138,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0046","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":116,\"b\":144,\"d\":5}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(116, 144, 5, 否, 否, 否, 4, 否)","explanation_zh":"擴展基礎核對=(116, 144, 5, 否, 否, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183046,"parameters":{"a":116,"b":144,"d":5,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":116,\"b\":144,\"d\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0047","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":120,\"b\":150,\"d\":30}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(120, 150, 30, 是, 是, 是, 30, 是)","explanation_zh":"擴展基礎核對=(120, 150, 30, 是, 是, 是, 30, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183047,"parameters":{"a":120,"b":150,"d":30,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":120,\"b\":150,\"d\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0048","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":124,\"b\":156,\"d\":1}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(124, 156, 1, 是, 是, 否, 4, 否)","explanation_zh":"擴展基礎核對=(124, 156, 1, 是, 是, 否, 4, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183048,"parameters":{"a":124,"b":156,"d":1,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":124,\"b\":156,\"d\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0049","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":128,\"b\":162,\"d\":3}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(128, 162, 3, 否, 是, 否, 2, 否)","explanation_zh":"擴展基礎核對=(128, 162, 3, 否, 是, 否, 2, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183049,"parameters":{"a":128,"b":162,"d":3,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":128,\"b\":162,\"d\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0050","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-common-divisor-audit；資料={\"a\":132,\"b\":168,\"d\":12}。請依 [a,b,d,d|a,d|b,d_is_gcd,gcd,valid] 的欄序回答。","answer":{"kind":"expression","value":"(132, 168, 12, 是, 是, 是, 12, 是)","explanation_zh":"擴展基礎核對=(132, 168, 12, 是, 是, 是, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-common-divisor-audit","seed":202610183050,"parameters":{"a":132,"b":168,"d":12,"domain":"Nat_gcd","operation":"gcd-common-divisor-audit","semantic_template":"qf-ext18-gcd-common-divisor-audit/{\"a\":132,\"b\":168,\"d\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0051","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":0,\"b\":1}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 是)","explanation_zh":"擴展基礎核對=(0, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183051,"parameters":{"a":0,"b":1,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":0,\"b\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0052","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":1,\"b\":3}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 是)","explanation_zh":"擴展基礎核對=(1, 3, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183052,"parameters":{"a":1,"b":3,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":1,\"b\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0053","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":2,\"b\":5}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(2, 5, 1, 是)","explanation_zh":"擴展基礎核對=(2, 5, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183053,"parameters":{"a":2,"b":5,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":2,\"b\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0054","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":3,\"b\":7}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(3, 7, 1, 是)","explanation_zh":"擴展基礎核對=(3, 7, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183054,"parameters":{"a":3,"b":7,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":3,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0055","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":4,\"b\":9}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(4, 9, 1, 是)","explanation_zh":"擴展基礎核對=(4, 9, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183055,"parameters":{"a":4,"b":9,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":4,\"b\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0056","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":5,\"b\":11}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(5, 11, 1, 是)","explanation_zh":"擴展基礎核對=(5, 11, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183056,"parameters":{"a":5,"b":11,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":5,\"b\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0057","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":6,\"b\":13}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(6, 13, 1, 是)","explanation_zh":"擴展基礎核對=(6, 13, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183057,"parameters":{"a":6,"b":13,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":6,\"b\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0058","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":7,\"b\":15}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(7, 15, 1, 是)","explanation_zh":"擴展基礎核對=(7, 15, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183058,"parameters":{"a":7,"b":15,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":7,\"b\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0059","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":8,\"b\":17}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(8, 17, 1, 是)","explanation_zh":"擴展基礎核對=(8, 17, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183059,"parameters":{"a":8,"b":17,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":8,\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0060","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":9,\"b\":19}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(9, 19, 1, 是)","explanation_zh":"擴展基礎核對=(9, 19, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183060,"parameters":{"a":9,"b":19,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":9,\"b\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0061","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":10,\"b\":21}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(10, 21, 1, 是)","explanation_zh":"擴展基礎核對=(10, 21, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183061,"parameters":{"a":10,"b":21,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":10,\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0062","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":11,\"b\":23}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(11, 23, 1, 是)","explanation_zh":"擴展基礎核對=(11, 23, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183062,"parameters":{"a":11,"b":23,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":11,\"b\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0063","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":12,\"b\":25}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(12, 25, 1, 是)","explanation_zh":"擴展基礎核對=(12, 25, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183063,"parameters":{"a":12,"b":25,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":12,\"b\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0064","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":13,\"b\":27}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(13, 27, 1, 是)","explanation_zh":"擴展基礎核對=(13, 27, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183064,"parameters":{"a":13,"b":27,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":13,\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0065","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":14,\"b\":29}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(14, 29, 1, 是)","explanation_zh":"擴展基礎核對=(14, 29, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183065,"parameters":{"a":14,"b":29,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":14,\"b\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0066","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":15,\"b\":31}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(15, 31, 1, 是)","explanation_zh":"擴展基礎核對=(15, 31, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183066,"parameters":{"a":15,"b":31,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":15,\"b\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0067","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":16,\"b\":33}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(16, 33, 1, 是)","explanation_zh":"擴展基礎核對=(16, 33, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183067,"parameters":{"a":16,"b":33,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":16,\"b\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0068","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":17,\"b\":35}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(17, 35, 1, 是)","explanation_zh":"擴展基礎核對=(17, 35, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183068,"parameters":{"a":17,"b":35,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":17,\"b\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0069","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":18,\"b\":37}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(18, 37, 1, 是)","explanation_zh":"擴展基礎核對=(18, 37, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183069,"parameters":{"a":18,"b":37,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":18,\"b\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0070","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":19,\"b\":39}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(19, 39, 1, 是)","explanation_zh":"擴展基礎核對=(19, 39, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183070,"parameters":{"a":19,"b":39,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":19,\"b\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0071","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":20,\"b\":41}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(20, 41, 1, 是)","explanation_zh":"擴展基礎核對=(20, 41, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183071,"parameters":{"a":20,"b":41,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":20,\"b\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0072","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":21,\"b\":43}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(21, 43, 1, 是)","explanation_zh":"擴展基礎核對=(21, 43, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183072,"parameters":{"a":21,"b":43,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":21,\"b\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0073","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":22,\"b\":45}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(22, 45, 1, 是)","explanation_zh":"擴展基礎核對=(22, 45, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183073,"parameters":{"a":22,"b":45,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":22,\"b\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0074","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":23,\"b\":47}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(23, 47, 1, 是)","explanation_zh":"擴展基礎核對=(23, 47, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183074,"parameters":{"a":23,"b":47,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":23,\"b\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0075","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-coprime-classification；資料={\"a\":24,\"b\":49}。請依 [a,b,gcd,coprime] 的欄序回答。","answer":{"kind":"expression","value":"(24, 49, 1, 是)","explanation_zh":"擴展基礎核對=(24, 49, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-coprime-classification","seed":202610183075,"parameters":{"a":24,"b":49,"domain":"Nat_gcd","operation":"gcd-coprime-classification","semantic_template":"qf-ext18-gcd-coprime-classification/{\"a\":24,\"b\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0076","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":12,\"b\":18,\"c\":30}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(12, 18, 30, 6, 6, 是)","explanation_zh":"擴展基礎核對=(12, 18, 30, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183076,"parameters":{"a":12,"b":18,"c":30,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":12,\"b\":18,\"c\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0077","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":13,\"b\":20,\"c\":33}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(13, 20, 33, 1, 1, 是)","explanation_zh":"擴展基礎核對=(13, 20, 33, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183077,"parameters":{"a":13,"b":20,"c":33,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":13,\"b\":20,\"c\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0078","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":14,\"b\":22,\"c\":36}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(14, 22, 36, 2, 2, 是)","explanation_zh":"擴展基礎核對=(14, 22, 36, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183078,"parameters":{"a":14,"b":22,"c":36,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":14,\"b\":22,\"c\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0079","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":15,\"b\":24,\"c\":39}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(15, 24, 39, 3, 3, 是)","explanation_zh":"擴展基礎核對=(15, 24, 39, 3, 3, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183079,"parameters":{"a":15,"b":24,"c":39,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":15,\"b\":24,\"c\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0080","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":16,\"b\":26,\"c\":42}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(16, 26, 42, 2, 2, 是)","explanation_zh":"擴展基礎核對=(16, 26, 42, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183080,"parameters":{"a":16,"b":26,"c":42,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":16,\"b\":26,\"c\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0081","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":17,\"b\":28,\"c\":45}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(17, 28, 45, 1, 1, 是)","explanation_zh":"擴展基礎核對=(17, 28, 45, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183081,"parameters":{"a":17,"b":28,"c":45,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":17,\"b\":28,\"c\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0082","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":18,\"b\":30,\"c\":48}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(18, 30, 48, 6, 6, 是)","explanation_zh":"擴展基礎核對=(18, 30, 48, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183082,"parameters":{"a":18,"b":30,"c":48,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":18,\"b\":30,\"c\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0083","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":19,\"b\":32,\"c\":51}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(19, 32, 51, 1, 1, 是)","explanation_zh":"擴展基礎核對=(19, 32, 51, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183083,"parameters":{"a":19,"b":32,"c":51,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":19,\"b\":32,\"c\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0084","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":20,\"b\":34,\"c\":54}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(20, 34, 54, 2, 2, 是)","explanation_zh":"擴展基礎核對=(20, 34, 54, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183084,"parameters":{"a":20,"b":34,"c":54,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":20,\"b\":34,\"c\":54}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0085","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":21,\"b\":36,\"c\":57}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(21, 36, 57, 3, 3, 是)","explanation_zh":"擴展基礎核對=(21, 36, 57, 3, 3, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183085,"parameters":{"a":21,"b":36,"c":57,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":21,\"b\":36,\"c\":57}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0086","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":22,\"b\":38,\"c\":60}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(22, 38, 60, 2, 2, 是)","explanation_zh":"擴展基礎核對=(22, 38, 60, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183086,"parameters":{"a":22,"b":38,"c":60,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":22,\"b\":38,\"c\":60}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0087","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":23,\"b\":40,\"c\":63}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(23, 40, 63, 1, 1, 是)","explanation_zh":"擴展基礎核對=(23, 40, 63, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183087,"parameters":{"a":23,"b":40,"c":63,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":23,\"b\":40,\"c\":63}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0088","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":24,\"b\":42,\"c\":66}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(24, 42, 66, 6, 6, 是)","explanation_zh":"擴展基礎核對=(24, 42, 66, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183088,"parameters":{"a":24,"b":42,"c":66,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":24,\"b\":42,\"c\":66}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0089","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":25,\"b\":44,\"c\":69}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(25, 44, 69, 1, 1, 是)","explanation_zh":"擴展基礎核對=(25, 44, 69, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183089,"parameters":{"a":25,"b":44,"c":69,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":25,\"b\":44,\"c\":69}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0090","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":26,\"b\":46,\"c\":72}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(26, 46, 72, 2, 2, 是)","explanation_zh":"擴展基礎核對=(26, 46, 72, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183090,"parameters":{"a":26,"b":46,"c":72,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":26,\"b\":46,\"c\":72}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0091","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":27,\"b\":48,\"c\":75}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(27, 48, 75, 3, 3, 是)","explanation_zh":"擴展基礎核對=(27, 48, 75, 3, 3, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183091,"parameters":{"a":27,"b":48,"c":75,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":27,\"b\":48,\"c\":75}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0092","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":28,\"b\":50,\"c\":78}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(28, 50, 78, 2, 2, 是)","explanation_zh":"擴展基礎核對=(28, 50, 78, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183092,"parameters":{"a":28,"b":50,"c":78,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":28,\"b\":50,\"c\":78}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0093","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":29,\"b\":52,\"c\":81}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(29, 52, 81, 1, 1, 是)","explanation_zh":"擴展基礎核對=(29, 52, 81, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183093,"parameters":{"a":29,"b":52,"c":81,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":29,\"b\":52,\"c\":81}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0094","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":30,\"b\":54,\"c\":84}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(30, 54, 84, 6, 6, 是)","explanation_zh":"擴展基礎核對=(30, 54, 84, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183094,"parameters":{"a":30,"b":54,"c":84,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":30,\"b\":54,\"c\":84}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0095","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":31,\"b\":56,\"c\":87}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(31, 56, 87, 1, 1, 是)","explanation_zh":"擴展基礎核對=(31, 56, 87, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183095,"parameters":{"a":31,"b":56,"c":87,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":31,\"b\":56,\"c\":87}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0096","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":32,\"b\":58,\"c\":90}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(32, 58, 90, 2, 2, 是)","explanation_zh":"擴展基礎核對=(32, 58, 90, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183096,"parameters":{"a":32,"b":58,"c":90,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":32,\"b\":58,\"c\":90}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0097","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":33,\"b\":60,\"c\":93}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(33, 60, 93, 3, 3, 是)","explanation_zh":"擴展基礎核對=(33, 60, 93, 3, 3, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183097,"parameters":{"a":33,"b":60,"c":93,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":33,\"b\":60,\"c\":93}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0098","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":34,\"b\":62,\"c\":96}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(34, 62, 96, 2, 2, 是)","explanation_zh":"擴展基礎核對=(34, 62, 96, 2, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183098,"parameters":{"a":34,"b":62,"c":96,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":34,\"b\":62,\"c\":96}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0099","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":35,\"b\":64,\"c\":99}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(35, 64, 99, 1, 1, 是)","explanation_zh":"擴展基礎核對=(35, 64, 99, 1, 1, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183099,"parameters":{"a":35,"b":64,"c":99,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":35,\"b\":64,\"c\":99}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r03-gcd-coprime-0100","batch_id":"batch-20261013-advance-20261018-r03-gcd-coprime-foundations","language":"zh-Hant","source_mko_ids":["mko-gcd-coprime"],"title_zh":"最大公因數與互質","stem_zh":"自然數gcd。操作=gcd-associative-triple；資料={\"a\":36,\"b\":66,\"c\":102}。請依 [a,b,c,left_gcd,right_gcd,equal] 的欄序回答。","answer":{"kind":"expression","value":"(36, 66, 102, 6, 6, 是)","explanation_zh":"擴展基礎核對=(36, 66, 102, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["最大公因數與互質"]},"variant":{"family_id":"qf-ext18-gcd-associative-triple","seed":202610183100,"parameters":{"a":36,"b":66,"c":102,"domain":"Nat_gcd","operation":"gcd-associative-triple","semantic_template":"qf-ext18-gcd-associative-triple/{\"a\":36,\"b\":66,\"c\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}