{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0001","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,5],\"residues\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], [0, 1], 6, 15, 是)","explanation_zh":"Wave5第四層核對=([3, 5], [0, 1], 6, 15, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045001,"parameters":{"moduli":[3,5],"residues":[0,1],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,5],\"residues\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0002","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,7],\"residues\":[1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 7], [1, 3], 10, 21, 是)","explanation_zh":"Wave5第四層核對=([3, 7], [1, 3], 10, 21, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045002,"parameters":{"moduli":[3,7],"residues":[1,3],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,7],\"residues\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0003","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,8],\"residues\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 8], [2, 5], 5, 24, 是)","explanation_zh":"Wave5第四層核對=([3, 8], [2, 5], 5, 24, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045003,"parameters":{"moduli":[3,8],"residues":[2,5],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,8],\"residues\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0004","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,10],\"residues\":[3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 10], [0, 7], 27, 30, 是)","explanation_zh":"Wave5第四層核對=([3, 10], [0, 7], 27, 30, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045004,"parameters":{"moduli":[3,10],"residues":[3,7],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,10],\"residues\":[3,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0005","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,11],\"residues\":[4,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 11], [1, 9], 31, 33, 是)","explanation_zh":"Wave5第四層核對=([3, 11], [1, 9], 31, 33, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045005,"parameters":{"moduli":[3,11],"residues":[4,9],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,11],\"residues\":[4,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0006","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,13],\"residues\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 13], [2, 11], 11, 39, 是)","explanation_zh":"Wave5第四層核對=([3, 13], [2, 11], 11, 39, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045006,"parameters":{"moduli":[3,13],"residues":[5,11],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,13],\"residues\":[5,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0007","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,14],\"residues\":[6,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 14], [0, 13], 27, 42, 是)","explanation_zh":"Wave5第四層核對=([3, 14], [0, 13], 27, 42, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045007,"parameters":{"moduli":[3,14],"residues":[6,13],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,14],\"residues\":[6,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0008","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,16],\"residues\":[7,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 16], [1, 15], 31, 48, 是)","explanation_zh":"Wave5第四層核對=([3, 16], [1, 15], 31, 48, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045008,"parameters":{"moduli":[3,16],"residues":[7,15],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,16],\"residues\":[7,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0009","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,17],\"residues\":[8,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 17], [2, 0], 17, 51, 是)","explanation_zh":"Wave5第四層核對=([3, 17], [2, 0], 17, 51, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045009,"parameters":{"moduli":[3,17],"residues":[8,17],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,17],\"residues\":[8,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0010","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,19],\"residues\":[9,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 19], [0, 0], 0, 57, 是)","explanation_zh":"Wave5第四層核對=([3, 19], [0, 0], 0, 57, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045010,"parameters":{"moduli":[3,19],"residues":[9,19],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,19],\"residues\":[9,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0011","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,20],\"residues\":[10,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 20], [1, 1], 1, 60, 是)","explanation_zh":"Wave5第四層核對=([3, 20], [1, 1], 1, 60, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045011,"parameters":{"moduli":[3,20],"residues":[10,21],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,20],\"residues\":[10,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0012","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,22],\"residues\":[11,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 22], [2, 1], 23, 66, 是)","explanation_zh":"Wave5第四層核對=([3, 22], [2, 1], 23, 66, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045012,"parameters":{"moduli":[3,22],"residues":[11,23],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,22],\"residues\":[11,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0013","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,23],\"residues\":[12,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 23], [0, 2], 48, 69, 是)","explanation_zh":"Wave5第四層核對=([3, 23], [0, 2], 48, 69, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045013,"parameters":{"moduli":[3,23],"residues":[12,25],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,23],\"residues\":[12,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0014","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,25],\"residues\":[13,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 25], [1, 2], 52, 75, 是)","explanation_zh":"Wave5第四層核對=([3, 25], [1, 2], 52, 75, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045014,"parameters":{"moduli":[3,25],"residues":[13,27],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,25],\"residues\":[13,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0015","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,26],\"residues\":[14,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 26], [2, 3], 29, 78, 是)","explanation_zh":"Wave5第四層核對=([3, 26], [2, 3], 29, 78, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045015,"parameters":{"moduli":[3,26],"residues":[14,29],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,26],\"residues\":[14,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0016","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,28],\"residues\":[15,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 28], [0, 3], 3, 84, 是)","explanation_zh":"Wave5第四層核對=([3, 28], [0, 3], 3, 84, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045016,"parameters":{"moduli":[3,28],"residues":[15,31],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,28],\"residues\":[15,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0017","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,29],\"residues\":[16,33]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 29], [1, 4], 4, 87, 是)","explanation_zh":"Wave5第四層核對=([3, 29], [1, 4], 4, 87, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045017,"parameters":{"moduli":[3,29],"residues":[16,33],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,29],\"residues\":[16,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0018","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,31],\"residues\":[17,35]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 31], [2, 4], 35, 93, 是)","explanation_zh":"Wave5第四層核對=([3, 31], [2, 4], 35, 93, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045018,"parameters":{"moduli":[3,31],"residues":[17,35],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,31],\"residues\":[17,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0019","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,32],\"residues\":[18,37]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 32], [0, 5], 69, 96, 是)","explanation_zh":"Wave5第四層核對=([3, 32], [0, 5], 69, 96, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045019,"parameters":{"moduli":[3,32],"residues":[18,37],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,32],\"residues\":[18,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0020","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,34],\"residues\":[19,39]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 34], [1, 5], 73, 102, 是)","explanation_zh":"Wave5第四層核對=([3, 34], [1, 5], 73, 102, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045020,"parameters":{"moduli":[3,34],"residues":[19,39],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,34],\"residues\":[19,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0021","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,35],\"residues\":[20,41]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 35], [2, 6], 41, 105, 是)","explanation_zh":"Wave5第四層核對=([3, 35], [2, 6], 41, 105, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045021,"parameters":{"moduli":[3,35],"residues":[20,41],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,35],\"residues\":[20,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0022","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,37],\"residues\":[21,43]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 37], [0, 6], 6, 111, 是)","explanation_zh":"Wave5第四層核對=([3, 37], [0, 6], 6, 111, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045022,"parameters":{"moduli":[3,37],"residues":[21,43],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,37],\"residues\":[21,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0023","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,38],\"residues\":[22,45]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 38], [1, 7], 7, 114, 是)","explanation_zh":"Wave5第四層核對=([3, 38], [1, 7], 7, 114, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045023,"parameters":{"moduli":[3,38],"residues":[22,45],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,38],\"residues\":[22,45]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0024","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,40],\"residues\":[23,47]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 40], [2, 7], 47, 120, 是)","explanation_zh":"Wave5第四層核對=([3, 40], [2, 7], 47, 120, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045024,"parameters":{"moduli":[3,40],"residues":[23,47],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,40],\"residues\":[23,47]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0025","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-pair；資料={\"moduli\":[3,41],\"residues\":[24,49]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 41], [0, 8], 90, 123, 是)","explanation_zh":"Wave5第四層核對=([3, 41], [0, 8], 90, 123, 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-pair","seed":202611045025,"parameters":{"moduli":[3,41],"residues":[24,49],"domain":"Exact_integer_CRT_system","operation":"crt-solve-pair","semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,41],\"residues\":[24,49]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0026","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,7],\"residues\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 7], [0, 1, 2], 51, 105, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 7], [0, 1, 2], 51, 105, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045026,"parameters":{"moduli":[3,5,7],"residues":[0,1,2],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,7],\"residues\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0027","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,8],\"residues\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 8], [1, 3, 5], 13, 120, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 8], [1, 3, 5], 13, 120, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045027,"parameters":{"moduli":[3,5,8],"residues":[1,3,5],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,8],\"residues\":[1,3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0028","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,11],\"residues\":[2,5,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 11], [2, 0, 8], 140, 165, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 11], [2, 0, 8], 140, 165, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045028,"parameters":{"moduli":[3,5,11],"residues":[2,5,8],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,11],\"residues\":[2,5,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0029","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,13],\"residues\":[3,7,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 13], [0, 2, 11], 102, 195, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 13], [0, 2, 11], 102, 195, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045029,"parameters":{"moduli":[3,5,13],"residues":[3,7,11],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,13],\"residues\":[3,7,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0030","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,14],\"residues\":[4,9,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 14], [1, 4, 0], 154, 210, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 14], [1, 4, 0], 154, 210, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045030,"parameters":{"moduli":[3,5,14],"residues":[4,9,14],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,14],\"residues\":[4,9,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0031","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,16],\"residues\":[5,11,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 16], [2, 1, 1], 161, 240, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 16], [2, 1, 1], 161, 240, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045031,"parameters":{"moduli":[3,5,16],"residues":[5,11,17],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,16],\"residues\":[5,11,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0032","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,17],\"residues\":[6,13,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 17], [0, 3, 3], 3, 255, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 17], [0, 3, 3], 3, 255, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045032,"parameters":{"moduli":[3,5,17],"residues":[6,13,20],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,17],\"residues\":[6,13,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0033","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,19],\"residues\":[7,15,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 19], [1, 0, 4], 175, 285, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 19], [1, 0, 4], 175, 285, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045033,"parameters":{"moduli":[3,5,19],"residues":[7,15,23],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,19],\"residues\":[7,15,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0034","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,22],\"residues\":[8,17,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 22], [2, 2, 4], 92, 330, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 22], [2, 2, 4], 92, 330, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045034,"parameters":{"moduli":[3,5,22],"residues":[8,17,26],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,22],\"residues\":[8,17,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0035","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,23],\"residues\":[9,19,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 23], [0, 4, 6], 144, 345, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 23], [0, 4, 6], 144, 345, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045035,"parameters":{"moduli":[3,5,23],"residues":[9,19,29],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,23],\"residues\":[9,19,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0036","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,26],\"residues\":[10,21,32]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 26], [1, 1, 6], 136, 390, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 26], [1, 1, 6], 136, 390, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045036,"parameters":{"moduli":[3,5,26],"residues":[10,21,32],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,26],\"residues\":[10,21,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0037","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,28],\"residues\":[11,23,35]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 28], [2, 3, 7], 203, 420, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 28], [2, 3, 7], 203, 420, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045037,"parameters":{"moduli":[3,5,28],"residues":[11,23,35],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,28],\"residues\":[11,23,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0038","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,29],\"residues\":[12,25,38]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 29], [0, 0, 9], 270, 435, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 29], [0, 0, 9], 270, 435, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045038,"parameters":{"moduli":[3,5,29],"residues":[12,25,38],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,29],\"residues\":[12,25,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0039","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,31],\"residues\":[13,27,41]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 31], [1, 2, 10], 382, 465, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 31], [1, 2, 10], 382, 465, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045039,"parameters":{"moduli":[3,5,31],"residues":[13,27,41],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,31],\"residues\":[13,27,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0040","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,32],\"residues\":[14,29,44]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 32], [2, 4, 12], 44, 480, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 32], [2, 4, 12], 44, 480, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045040,"parameters":{"moduli":[3,5,32],"residues":[14,29,44],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,32],\"residues\":[14,29,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0041","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,34],\"residues\":[15,31,47]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 34], [0, 1, 13], 81, 510, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 34], [0, 1, 13], 81, 510, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045041,"parameters":{"moduli":[3,5,34],"residues":[15,31,47],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,34],\"residues\":[15,31,47]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0042","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,37],\"residues\":[16,33,50]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 37], [1, 3, 13], 13, 555, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 37], [1, 3, 13], 13, 555, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045042,"parameters":{"moduli":[3,5,37],"residues":[16,33,50],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,37],\"residues\":[16,33,50]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0043","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,38],\"residues\":[17,35,53]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 38], [2, 0, 15], 395, 570, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 38], [2, 0, 15], 395, 570, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045043,"parameters":{"moduli":[3,5,38],"residues":[17,35,53],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,38],\"residues\":[17,35,53]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0044","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,41],\"residues\":[18,37,56]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 41], [0, 2, 15], 507, 615, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 41], [0, 2, 15], 507, 615, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045044,"parameters":{"moduli":[3,5,41],"residues":[18,37,56],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,41],\"residues\":[18,37,56]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0045","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,43],\"residues\":[19,39,59]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 43], [1, 4, 16], 274, 645, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 43], [1, 4, 16], 274, 645, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045045,"parameters":{"moduli":[3,5,43],"residues":[19,39,59],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,43],\"residues\":[19,39,59]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0046","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,44],\"residues\":[20,41,62]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 44], [2, 1, 18], 326, 660, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 44], [2, 1, 18], 326, 660, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045046,"parameters":{"moduli":[3,5,44],"residues":[20,41,62],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,44],\"residues\":[20,41,62]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0047","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,46],\"residues\":[21,43,65]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 46], [0, 3, 19], 663, 690, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 46], [0, 3, 19], 663, 690, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045047,"parameters":{"moduli":[3,5,46],"residues":[21,43,65],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,46],\"residues\":[21,43,65]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0048","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,47],\"residues\":[22,45,68]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 47], [1, 0, 21], 115, 705, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 47], [1, 0, 21], 115, 705, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045048,"parameters":{"moduli":[3,5,47],"residues":[22,45,68],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,47],\"residues\":[22,45,68]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0049","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,49],\"residues\":[23,47,71]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 49], [2, 2, 22], 512, 735, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 49], [2, 2, 22], 512, 735, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045049,"parameters":{"moduli":[3,5,49],"residues":[23,47,71],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,49],\"residues\":[23,47,71]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0050","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={\"moduli\":[3,5,52],\"residues\":[24,49,74]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 52], [0, 4, 22], 594, 780, [是, 是, 是], 是)","explanation_zh":"Wave5第四層核對=([3, 5, 52], [0, 4, 22], 594, 780, [是, 是, 是], 是)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-solve-system","seed":202611045050,"parameters":{"moduli":[3,5,52],"residues":[24,49,74],"domain":"Exact_integer_CRT_system","operation":"crt-solve-system","semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,52],\"residues\":[24,49,74]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0051","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,5],\"residues\":[0,1],\"other\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 21, 15, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(6, 21, 15, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045051,"parameters":{"moduli":[3,5],"residues":[0,1],"other":21,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,5],\"other\":21,\"residues\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0052","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,7],\"residues\":[1,3],\"other\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 31, 21, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(10, 31, 21, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045052,"parameters":{"moduli":[3,7],"residues":[1,3],"other":31,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,7],\"other\":31,\"residues\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0053","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,8],\"residues\":[2,5],\"other\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 29, 24, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(5, 29, 24, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045053,"parameters":{"moduli":[3,8],"residues":[2,5],"other":29,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,8],\"other\":29,\"residues\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0054","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,10],\"residues\":[3,7],\"other\":57}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 57, 30, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(27, 57, 30, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045054,"parameters":{"moduli":[3,10],"residues":[3,7],"other":57,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,10],\"other\":57,\"residues\":[3,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0055","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,11],\"residues\":[4,9],\"other\":64}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 64, 33, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(31, 64, 33, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045055,"parameters":{"moduli":[3,11],"residues":[4,9],"other":64,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,11],\"other\":64,\"residues\":[4,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0056","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,13],\"residues\":[5,11],\"other\":50}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 50, 39, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(11, 50, 39, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045056,"parameters":{"moduli":[3,13],"residues":[5,11],"other":50,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,13],\"other\":50,\"residues\":[5,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0057","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,14],\"residues\":[6,13],\"other\":69}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 69, 42, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(27, 69, 42, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045057,"parameters":{"moduli":[3,14],"residues":[6,13],"other":69,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,14],\"other\":69,\"residues\":[6,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0058","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,16],\"residues\":[7,15],\"other\":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 79, 48, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(31, 79, 48, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045058,"parameters":{"moduli":[3,16],"residues":[7,15],"other":79,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,16],\"other\":79,\"residues\":[7,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0059","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,17],\"residues\":[8,17],\"other\":68}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 68, 51, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(17, 68, 51, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045059,"parameters":{"moduli":[3,17],"residues":[8,17],"other":68,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,17],\"other\":68,\"residues\":[8,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0060","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,19],\"residues\":[9,19],\"other\":57}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 57, 57, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(0, 57, 57, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045060,"parameters":{"moduli":[3,19],"residues":[9,19],"other":57,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,19],\"other\":57,\"residues\":[9,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0061","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,20],\"residues\":[10,21],\"other\":61}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 61, 60, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(1, 61, 60, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045061,"parameters":{"moduli":[3,20],"residues":[10,21],"other":61,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,20],\"other\":61,\"residues\":[10,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0062","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,22],\"residues\":[11,23],\"other\":89}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 89, 66, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(23, 89, 66, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045062,"parameters":{"moduli":[3,22],"residues":[11,23],"other":89,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,22],\"other\":89,\"residues\":[11,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0063","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,23],\"residues\":[12,25],\"other\":117}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(48, 117, 69, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(48, 117, 69, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045063,"parameters":{"moduli":[3,23],"residues":[12,25],"other":117,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,23],\"other\":117,\"residues\":[12,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0064","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,25],\"residues\":[13,27],\"other\":127}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(52, 127, 75, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(52, 127, 75, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045064,"parameters":{"moduli":[3,25],"residues":[13,27],"other":127,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,25],\"other\":127,\"residues\":[13,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0065","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,26],\"residues\":[14,29],\"other\":107}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 107, 78, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(29, 107, 78, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045065,"parameters":{"moduli":[3,26],"residues":[14,29],"other":107,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,26],\"other\":107,\"residues\":[14,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0066","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,28],\"residues\":[15,31],\"other\":87}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 87, 84, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(3, 87, 84, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045066,"parameters":{"moduli":[3,28],"residues":[15,31],"other":87,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,28],\"other\":87,\"residues\":[15,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0067","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,29],\"residues\":[16,33],\"other\":91}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 91, 87, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(4, 91, 87, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045067,"parameters":{"moduli":[3,29],"residues":[16,33],"other":91,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,29],\"other\":91,\"residues\":[16,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0068","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,31],\"residues\":[17,35],\"other\":128}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(35, 128, 93, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(35, 128, 93, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045068,"parameters":{"moduli":[3,31],"residues":[17,35],"other":128,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,31],\"other\":128,\"residues\":[17,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0069","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,32],\"residues\":[18,37],\"other\":165}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(69, 165, 96, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(69, 165, 96, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045069,"parameters":{"moduli":[3,32],"residues":[18,37],"other":165,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,32],\"other\":165,\"residues\":[18,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0070","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,34],\"residues\":[19,39],\"other\":175}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 175, 102, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(73, 175, 102, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045070,"parameters":{"moduli":[3,34],"residues":[19,39],"other":175,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,34],\"other\":175,\"residues\":[19,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0071","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,35],\"residues\":[20,41],\"other\":146}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 146, 105, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(41, 146, 105, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045071,"parameters":{"moduli":[3,35],"residues":[20,41],"other":146,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,35],\"other\":146,\"residues\":[20,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0072","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,37],\"residues\":[21,43],\"other\":117}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 117, 111, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(6, 117, 111, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045072,"parameters":{"moduli":[3,37],"residues":[21,43],"other":117,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,37],\"other\":117,\"residues\":[21,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0073","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,38],\"residues\":[22,45],\"other\":121}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 121, 114, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(7, 121, 114, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045073,"parameters":{"moduli":[3,38],"residues":[22,45],"other":121,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,38],\"other\":121,\"residues\":[22,45]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0074","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,40],\"residues\":[23,47],\"other\":167}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 167, 120, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(47, 167, 120, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045074,"parameters":{"moduli":[3,40],"residues":[23,47],"other":167,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,40],\"other\":167,\"residues\":[23,47]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0075","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={\"moduli\":[3,41],\"residues\":[24,49],\"other\":213}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(90, 213, 123, 是, unique_modulo_product_not_unique_integer)","explanation_zh":"Wave5第四層核對=(90, 213, 123, 是, unique_modulo_product_not_unique_integer)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045075,"parameters":{"moduli":[3,41],"residues":[24,49],"other":213,"domain":"Exact_integer_CRT_system","operation":"crt-uniqueness-mod-product","semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,41],\"other\":213,\"residues\":[24,49]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0076","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[4,6],\"residues\":[1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 6], [1, 3], 是, 9, 12, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([4, 6], [1, 3], 是, 9, 12, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045076,"parameters":{"moduli":[4,6],"residues":[1,3],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[4,6],\"residues\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0077","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[6,9],\"residues\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 9], [1, 4], 是, 13, 18, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([6, 9], [1, 4], 是, 13, 18, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045077,"parameters":{"moduli":[6,9],"residues":[1,4],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[6,9],\"residues\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0078","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[8,12],\"residues\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 12], [1, 5], 是, 17, 24, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([8, 12], [1, 5], 是, 17, 24, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045078,"parameters":{"moduli":[8,12],"residues":[1,5],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[8,12],\"residues\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0079","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[10,15],\"residues\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 15], [1, 6], 是, 21, 30, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([10, 15], [1, 6], 是, 21, 30, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045079,"parameters":{"moduli":[10,15],"residues":[1,6],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[10,15],\"residues\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0080","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[12,18],\"residues\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 18], [1, 7], 是, 25, 36, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([12, 18], [1, 7], 是, 25, 36, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045080,"parameters":{"moduli":[12,18],"residues":[1,7],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[12,18],\"residues\":[1,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0081","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[14,21],\"residues\":[1,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 21], [1, 8], 是, 29, 42, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([14, 21], [1, 8], 是, 29, 42, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045081,"parameters":{"moduli":[14,21],"residues":[1,8],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[14,21],\"residues\":[1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0082","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[16,24],\"residues\":[1,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 24], [1, 9], 是, 33, 48, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([16, 24], [1, 9], 是, 33, 48, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045082,"parameters":{"moduli":[16,24],"residues":[1,9],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[16,24],\"residues\":[1,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0083","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[18,27],\"residues\":[1,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 27], [1, 10], 是, 37, 54, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([18, 27], [1, 10], 是, 37, 54, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045083,"parameters":{"moduli":[18,27],"residues":[1,10],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[18,27],\"residues\":[1,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0084","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[20,30],\"residues\":[1,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 30], [1, 11], 是, 41, 60, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([20, 30], [1, 11], 是, 41, 60, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045084,"parameters":{"moduli":[20,30],"residues":[1,11],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[20,30],\"residues\":[1,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0085","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[22,33],\"residues\":[1,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 33], [1, 12], 是, 45, 66, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([22, 33], [1, 12], 是, 45, 66, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045085,"parameters":{"moduli":[22,33],"residues":[1,12],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[22,33],\"residues\":[1,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0086","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[24,36],\"residues\":[1,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 36], [1, 13], 是, 49, 72, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([24, 36], [1, 13], 是, 49, 72, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045086,"parameters":{"moduli":[24,36],"residues":[1,13],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[24,36],\"residues\":[1,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0087","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[26,39],\"residues\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([26, 39], [1, 14], 是, 53, 78, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([26, 39], [1, 14], 是, 53, 78, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045087,"parameters":{"moduli":[26,39],"residues":[1,14],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[26,39],\"residues\":[1,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0088","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[28,42],\"residues\":[1,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([28, 42], [1, 15], 是, 57, 84, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([28, 42], [1, 15], 是, 57, 84, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045088,"parameters":{"moduli":[28,42],"residues":[1,15],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[28,42],\"residues\":[1,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0089","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[30,45],\"residues\":[1,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([30, 45], [1, 16], 是, 61, 90, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([30, 45], [1, 16], 是, 61, 90, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045089,"parameters":{"moduli":[30,45],"residues":[1,16],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[30,45],\"residues\":[1,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0090","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[32,48],\"residues\":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([32, 48], [1, 17], 是, 65, 96, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([32, 48], [1, 17], 是, 65, 96, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045090,"parameters":{"moduli":[32,48],"residues":[1,17],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[32,48],\"residues\":[1,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0091","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[34,51],\"residues\":[1,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([34, 51], [1, 18], 是, 69, 102, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([34, 51], [1, 18], 是, 69, 102, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045091,"parameters":{"moduli":[34,51],"residues":[1,18],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[34,51],\"residues\":[1,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0092","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[36,54],\"residues\":[1,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([36, 54], [1, 19], 是, 73, 108, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([36, 54], [1, 19], 是, 73, 108, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045092,"parameters":{"moduli":[36,54],"residues":[1,19],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[36,54],\"residues\":[1,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0093","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[38,57],\"residues\":[1,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([38, 57], [1, 20], 是, 77, 114, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([38, 57], [1, 20], 是, 77, 114, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045093,"parameters":{"moduli":[38,57],"residues":[1,20],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[38,57],\"residues\":[1,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0094","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[40,60],\"residues\":[1,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([40, 60], [1, 21], 是, 81, 120, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([40, 60], [1, 21], 是, 81, 120, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045094,"parameters":{"moduli":[40,60],"residues":[1,21],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[40,60],\"residues\":[1,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0095","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[42,63],\"residues\":[1,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([42, 63], [1, 22], 是, 85, 126, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([42, 63], [1, 22], 是, 85, 126, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045095,"parameters":{"moduli":[42,63],"residues":[1,22],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[42,63],\"residues\":[1,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0096","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[44,66],\"residues\":[1,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([44, 66], [1, 23], 是, 89, 132, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([44, 66], [1, 23], 是, 89, 132, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045096,"parameters":{"moduli":[44,66],"residues":[1,23],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[44,66],\"residues\":[1,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0097","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[46,69],\"residues\":[1,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([46, 69], [1, 24], 是, 93, 138, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([46, 69], [1, 24], 是, 93, 138, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045097,"parameters":{"moduli":[46,69],"residues":[1,24],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[46,69],\"residues\":[1,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0098","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[48,72],\"residues\":[1,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([48, 72], [1, 25], 是, 97, 144, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([48, 72], [1, 25], 是, 97, 144, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045098,"parameters":{"moduli":[48,72],"residues":[1,25],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[48,72],\"residues\":[1,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0099","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[50,75],\"residues\":[1,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([50, 75], [1, 26], 是, 101, 150, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([50, 75], [1, 26], 是, 101, 150, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045099,"parameters":{"moduli":[50,75],"residues":[1,26],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[50,75],\"residues\":[1,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r05-chinese-remainder-theorem-0100","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-chinese-remainder-theorem"],"title_zh":"中國剩餘定理","stem_zh":"正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[52,78],\"residues\":[1,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([52, 78], [1, 27], 是, 105, 156, noncoprime_not_automatically_inconsistent)","explanation_zh":"Wave5第四層核對=([52, 78], [1, 27], 是, 105, 156, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["中國剩餘定理"]},"variant":{"family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045100,"parameters":{"moduli":[52,78],"residues":[1,27],"domain":"Exact_integer_CRT_system","operation":"crt-noncoprime-boundary","semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[52,78],\"residues\":[1,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}