{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0001","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":27}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 27, 1, 是, 14)","explanation_zh":"Wave5第三層核對=(2, 27, 1, 是, 14)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037001,"parameters":{"a":2,"n":27,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0002","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":28}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 28, 1, 是, 19)","explanation_zh":"Wave5第三層核對=(3, 28, 1, 是, 19)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037002,"parameters":{"a":3,"n":28,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0003","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":29}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 29, 1, 是, 15)","explanation_zh":"Wave5第三層核對=(2, 29, 1, 是, 15)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037003,"parameters":{"a":2,"n":29,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0004","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":7,\"n\":30}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(7, 30, 1, 是, 13)","explanation_zh":"Wave5第三層核對=(7, 30, 1, 是, 13)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037004,"parameters":{"a":7,"n":30,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":7,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0005","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":31}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 31, 1, 是, 16)","explanation_zh":"Wave5第三層核對=(2, 31, 1, 是, 16)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037005,"parameters":{"a":2,"n":31,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0006","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":32}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 32, 1, 是, 11)","explanation_zh":"Wave5第三層核對=(3, 32, 1, 是, 11)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037006,"parameters":{"a":3,"n":32,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0007","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":33}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 33, 1, 是, 17)","explanation_zh":"Wave5第三層核對=(2, 33, 1, 是, 17)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037007,"parameters":{"a":2,"n":33,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0008","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":34}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 34, 1, 是, 23)","explanation_zh":"Wave5第三層核對=(3, 34, 1, 是, 23)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037008,"parameters":{"a":3,"n":34,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0009","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":35}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 35, 1, 是, 18)","explanation_zh":"Wave5第三層核對=(2, 35, 1, 是, 18)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037009,"parameters":{"a":2,"n":35,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0010","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":5,\"n\":36}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(5, 36, 1, 是, 29)","explanation_zh":"Wave5第三層核對=(5, 36, 1, 是, 29)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037010,"parameters":{"a":5,"n":36,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0011","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":37}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 37, 1, 是, 19)","explanation_zh":"Wave5第三層核對=(2, 37, 1, 是, 19)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037011,"parameters":{"a":2,"n":37,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0012","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":38}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 38, 1, 是, 13)","explanation_zh":"Wave5第三層核對=(3, 38, 1, 是, 13)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037012,"parameters":{"a":3,"n":38,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0013","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":39}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 39, 1, 是, 20)","explanation_zh":"Wave5第三層核對=(2, 39, 1, 是, 20)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037013,"parameters":{"a":2,"n":39,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0014","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":40}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 40, 1, 是, 27)","explanation_zh":"Wave5第三層核對=(3, 40, 1, 是, 27)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037014,"parameters":{"a":3,"n":40,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0015","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":41}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 41, 1, 是, 21)","explanation_zh":"Wave5第三層核對=(2, 41, 1, 是, 21)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037015,"parameters":{"a":2,"n":41,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0016","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":5,\"n\":42}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(5, 42, 1, 是, 17)","explanation_zh":"Wave5第三層核對=(5, 42, 1, 是, 17)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037016,"parameters":{"a":5,"n":42,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0017","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":43}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 43, 1, 是, 22)","explanation_zh":"Wave5第三層核對=(2, 43, 1, 是, 22)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037017,"parameters":{"a":2,"n":43,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0018","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":44}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 44, 1, 是, 15)","explanation_zh":"Wave5第三層核對=(3, 44, 1, 是, 15)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037018,"parameters":{"a":3,"n":44,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0019","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":45}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 45, 1, 是, 23)","explanation_zh":"Wave5第三層核對=(2, 45, 1, 是, 23)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037019,"parameters":{"a":2,"n":45,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0020","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":46}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 46, 1, 是, 31)","explanation_zh":"Wave5第三層核對=(3, 46, 1, 是, 31)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037020,"parameters":{"a":3,"n":46,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0021","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":47}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 47, 1, 是, 24)","explanation_zh":"Wave5第三層核對=(2, 47, 1, 是, 24)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037021,"parameters":{"a":2,"n":47,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0022","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":5,\"n\":48}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(5, 48, 1, 是, 29)","explanation_zh":"Wave5第三層核對=(5, 48, 1, 是, 29)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037022,"parameters":{"a":5,"n":48,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0023","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":49}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 49, 1, 是, 25)","explanation_zh":"Wave5第三層核對=(2, 49, 1, 是, 25)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037023,"parameters":{"a":2,"n":49,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0024","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":3,\"n\":50}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(3, 50, 1, 是, 17)","explanation_zh":"Wave5第三層核對=(3, 50, 1, 是, 17)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037024,"parameters":{"a":3,"n":50,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":50}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0025","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-existence；資料={\"a\":2,\"n\":51}。請依 [a,n,gcd,exists,inverse] 欄序回答。","answer":{"kind":"expression","value":"(2, 51, 1, 是, 26)","explanation_zh":"Wave5第三層核對=(2, 51, 1, 是, 26)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-existence","seed":202611037025,"parameters":{"a":2,"n":51,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-existence","semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0026","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":27}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 27, 1, -13, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 27, 1, -13, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037026,"parameters":{"a":2,"n":27,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0027","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":28}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 28, 1, -9, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 28, 1, -9, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037027,"parameters":{"a":3,"n":28,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0028","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":29}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 29, 1, -14, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 29, 1, -14, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037028,"parameters":{"a":2,"n":29,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0029","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":7,\"n\":30}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(7, 30, 1, 13, -3, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(7, 30, 1, 13, -3, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037029,"parameters":{"a":7,"n":30,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":7,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0030","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":31}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 31, 1, -15, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 31, 1, -15, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037030,"parameters":{"a":2,"n":31,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0031","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":32}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 32, 1, 11, -1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 32, 1, 11, -1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037031,"parameters":{"a":3,"n":32,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0032","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":33}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 33, 1, -16, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 33, 1, -16, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037032,"parameters":{"a":2,"n":33,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0033","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":34}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 34, 1, -11, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 34, 1, -11, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037033,"parameters":{"a":3,"n":34,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0034","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":35}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 35, 1, -17, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 35, 1, -17, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037034,"parameters":{"a":2,"n":35,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0035","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":5,\"n\":36}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 36, 1, -7, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(5, 36, 1, -7, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037035,"parameters":{"a":5,"n":36,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0036","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":37}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 37, 1, -18, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 37, 1, -18, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037036,"parameters":{"a":2,"n":37,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0037","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":38}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 38, 1, 13, -1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 38, 1, 13, -1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037037,"parameters":{"a":3,"n":38,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0038","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":39}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 39, 1, -19, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 39, 1, -19, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037038,"parameters":{"a":2,"n":39,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0039","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":40}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 40, 1, -13, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 40, 1, -13, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037039,"parameters":{"a":3,"n":40,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0040","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":41}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 41, 1, -20, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 41, 1, -20, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037040,"parameters":{"a":2,"n":41,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0041","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":5,\"n\":42}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 42, 1, 17, -2, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(5, 42, 1, 17, -2, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037041,"parameters":{"a":5,"n":42,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0042","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":43}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 43, 1, -21, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 43, 1, -21, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037042,"parameters":{"a":2,"n":43,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0043","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":44}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 44, 1, 15, -1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 44, 1, 15, -1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037043,"parameters":{"a":3,"n":44,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0044","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":45}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 45, 1, -22, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 45, 1, -22, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037044,"parameters":{"a":2,"n":45,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0045","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":46}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 46, 1, -15, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 46, 1, -15, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037045,"parameters":{"a":3,"n":46,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0046","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":47}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 47, 1, -23, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 47, 1, -23, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037046,"parameters":{"a":2,"n":47,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0047","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":5,\"n\":48}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 48, 1, -19, 2, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(5, 48, 1, -19, 2, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037047,"parameters":{"a":5,"n":48,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0048","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":49}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 49, 1, -24, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 49, 1, -24, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037048,"parameters":{"a":2,"n":49,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0049","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":3,\"n\":50}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 50, 1, 17, -1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(3, 50, 1, 17, -1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037049,"parameters":{"a":3,"n":50,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":50}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0050","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-extended-euclid；資料={\"a\":2,\"n\":51}。請依 [a,n,gcd,x,y,bezout_value,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 51, 1, -25, 1, 1, Bezout_equals_one)","explanation_zh":"Wave5第三層核對=(2, 51, 1, -25, 1, 1, Bezout_equals_one)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-extended-euclid","seed":202611037050,"parameters":{"a":2,"n":51,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-extended-euclid","semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0051","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":27}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 27, 14, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 27, 14, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037051,"parameters":{"a":2,"n":27,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0052","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":28}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 28, 19, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 28, 19, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037052,"parameters":{"a":3,"n":28,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0053","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":29}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 29, 15, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 29, 15, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037053,"parameters":{"a":2,"n":29,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0054","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":7,\"n\":30}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(7, 30, 13, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(7, 30, 13, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037054,"parameters":{"a":7,"n":30,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":7,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0055","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":31}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 31, 16, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 31, 16, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037055,"parameters":{"a":2,"n":31,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0056","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":32}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 32, 11, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 32, 11, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037056,"parameters":{"a":3,"n":32,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0057","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":33}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 33, 17, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 33, 17, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037057,"parameters":{"a":2,"n":33,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0058","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":34}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 34, 23, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 34, 23, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037058,"parameters":{"a":3,"n":34,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0059","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":35}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 35, 18, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 35, 18, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037059,"parameters":{"a":2,"n":35,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0060","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":5,\"n\":36}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(5, 36, 29, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(5, 36, 29, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037060,"parameters":{"a":5,"n":36,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0061","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":37}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 37, 19, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 37, 19, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037061,"parameters":{"a":2,"n":37,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0062","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":38}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 38, 13, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 38, 13, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037062,"parameters":{"a":3,"n":38,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0063","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":39}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 39, 20, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 39, 20, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037063,"parameters":{"a":2,"n":39,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0064","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":40}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 40, 27, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 40, 27, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037064,"parameters":{"a":3,"n":40,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0065","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":41}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 41, 21, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 41, 21, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037065,"parameters":{"a":2,"n":41,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0066","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":5,\"n\":42}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(5, 42, 17, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(5, 42, 17, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037066,"parameters":{"a":5,"n":42,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0067","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":43}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 43, 22, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 43, 22, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037067,"parameters":{"a":2,"n":43,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0068","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":44}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 44, 15, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 44, 15, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037068,"parameters":{"a":3,"n":44,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0069","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":45}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 45, 23, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 45, 23, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037069,"parameters":{"a":2,"n":45,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0070","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":46}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 46, 31, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 46, 31, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037070,"parameters":{"a":3,"n":46,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0071","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":47}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 47, 24, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 47, 24, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037071,"parameters":{"a":2,"n":47,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0072","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":5,\"n\":48}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(5, 48, 29, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(5, 48, 29, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037072,"parameters":{"a":5,"n":48,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0073","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":49}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 49, 25, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 49, 25, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037073,"parameters":{"a":2,"n":49,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0074","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":3,\"n\":50}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(3, 50, 17, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(3, 50, 17, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037074,"parameters":{"a":3,"n":50,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":50}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0075","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-normalized-inverse；資料={\"a\":2,\"n\":51}。請依 [a,n,inverse,product_mod_n,valid,uniqueness] 欄序回答。","answer":{"kind":"expression","value":"(2, 51, 26, 1, 是, unique_modulo_n)","explanation_zh":"Wave5第三層核對=(2, 51, 26, 1, 是, unique_modulo_n)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037075,"parameters":{"a":2,"n":51,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-normalized-inverse","semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0076","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":2,\"n\":4}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 2, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(2, 4, 2, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037076,"parameters":{"a":2,"n":4,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":2,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0077","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":3,\"n\":6}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 3, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(3, 6, 3, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037077,"parameters":{"a":3,"n":6,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0078","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":4,\"n\":8}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 4, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(4, 8, 4, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037078,"parameters":{"a":4,"n":8,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":4,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0079","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":5,\"n\":10}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 10, 5, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(5, 10, 5, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037079,"parameters":{"a":5,"n":10,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":5,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0080","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":6,\"n\":12}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 6, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(6, 12, 6, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037080,"parameters":{"a":6,"n":12,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":6,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0081","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":7,\"n\":14}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7, 14, 7, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(7, 14, 7, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037081,"parameters":{"a":7,"n":14,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":7,\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0082","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":8,\"n\":16}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8, 16, 8, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(8, 16, 8, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037082,"parameters":{"a":8,"n":16,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":8,\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0083","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":9,\"n\":18}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9, 18, 9, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(9, 18, 9, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037083,"parameters":{"a":9,"n":18,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":9,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0084","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":10,\"n\":20}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10, 20, 10, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(10, 20, 10, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037084,"parameters":{"a":10,"n":20,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":10,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0085","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":11,\"n\":22}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11, 22, 11, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(11, 22, 11, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037085,"parameters":{"a":11,"n":22,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":11,\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0086","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":12,\"n\":24}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12, 24, 12, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(12, 24, 12, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037086,"parameters":{"a":12,"n":24,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":12,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0087","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":13,\"n\":26}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13, 26, 13, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(13, 26, 13, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037087,"parameters":{"a":13,"n":26,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":13,\"n\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0088","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":14,\"n\":28}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14, 28, 14, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(14, 28, 14, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037088,"parameters":{"a":14,"n":28,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":14,\"n\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0089","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":15,\"n\":30}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15, 30, 15, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(15, 30, 15, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037089,"parameters":{"a":15,"n":30,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":15,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0090","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":16,\"n\":32}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16, 32, 16, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(16, 32, 16, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037090,"parameters":{"a":16,"n":32,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":16,\"n\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0091","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":17,\"n\":34}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17, 34, 17, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(17, 34, 17, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037091,"parameters":{"a":17,"n":34,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":17,\"n\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0092","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":18,\"n\":36}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18, 36, 18, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(18, 36, 18, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037092,"parameters":{"a":18,"n":36,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":18,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0093","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":19,\"n\":38}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19, 38, 19, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(19, 38, 19, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037093,"parameters":{"a":19,"n":38,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":19,\"n\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0094","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":20,\"n\":40}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20, 40, 20, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(20, 40, 20, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037094,"parameters":{"a":20,"n":40,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":20,\"n\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0095","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":21,\"n\":42}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21, 42, 21, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(21, 42, 21, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037095,"parameters":{"a":21,"n":42,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":21,\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0096","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":22,\"n\":44}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22, 44, 22, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(22, 44, 22, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037096,"parameters":{"a":22,"n":44,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":22,\"n\":44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0097","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":23,\"n\":46}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23, 46, 23, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(23, 46, 23, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037097,"parameters":{"a":23,"n":46,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":23,\"n\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0098","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":24,\"n\":48}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24, 48, 24, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(24, 48, 24, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037098,"parameters":{"a":24,"n":48,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":24,\"n\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0099","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":25,\"n\":50}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(25, 50, 25, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(25, 50, 25, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037099,"parameters":{"a":25,"n":50,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":25,\"n\":50}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r07-modular-inverse-0100","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-inverse"],"title_zh":"模反元素","stem_zh":"整數a與模數n大於等於2的exact Euclidean arithmetic。操作=modinv-nonunit-boundary；資料={\"a\":26,\"n\":52}。請依 [a,n,gcd,inverse,exists,boundary] 欄序回答。","answer":{"kind":"expression","value":"(26, 52, 26, null, 否, nonunit_has_no_modular_inverse)","explanation_zh":"Wave5第三層核對=(26, 52, 26, null, 否, nonunit_has_no_modular_inverse)；依完整明示資料重算；存在等價gcd等於1且extended Euclid給Bezout證書；輸出normalized residue且唯一只在mod n；nonunit無inverse。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模反元素"]},"variant":{"family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037100,"parameters":{"a":26,"n":52,"domain":"Exact_extended_Euclid_modular_inverse","operation":"modinv-nonunit-boundary","semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":26,\"n\":52}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}