{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0001","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":1}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(1, [1], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(1, [1], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016001,"parameters":{"n":1,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0002","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":2}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [1], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(2, [1], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016002,"parameters":{"n":2,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0003","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":3}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(3, [1, 2], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016003,"parameters":{"n":3,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0004","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":4}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 3], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(4, [1, 3], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016004,"parameters":{"n":4,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0005","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":5}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(5, [1, 2, 3, 4], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(5, [1, 2, 3, 4], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016005,"parameters":{"n":5,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0006","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":6}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 5], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(6, [1, 5], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016006,"parameters":{"n":6,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0007","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":7}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(7, [1, 2, 3, 4, 5, 6], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(7, [1, 2, 3, 4, 5, 6], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016007,"parameters":{"n":7,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0008","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":8}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(8, [1, 3, 5, 7], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(8, [1, 3, 5, 7], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016008,"parameters":{"n":8,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0009","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":9}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(9, [1, 2, 4, 5, 7, 8], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(9, [1, 2, 4, 5, 7, 8], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016009,"parameters":{"n":9,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0010","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":10}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(10, [1, 3, 7, 9], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(10, [1, 3, 7, 9], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016010,"parameters":{"n":10,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0011","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":11}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016011,"parameters":{"n":11,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0012","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":12}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 5, 7, 11], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(12, [1, 5, 7, 11], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016012,"parameters":{"n":12,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0013","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":13}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016013,"parameters":{"n":13,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0014","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":14}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(14, [1, 3, 5, 9, 11, 13], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(14, [1, 3, 5, 9, 11, 13], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016014,"parameters":{"n":14,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0015","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":15}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016015,"parameters":{"n":15,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0016","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":16}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016016,"parameters":{"n":16,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0017","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":17}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016017,"parameters":{"n":17,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0018","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":18}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(18, [1, 5, 7, 11, 13, 17], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(18, [1, 5, 7, 11, 13, 17], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016018,"parameters":{"n":18,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0019","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":19}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016019,"parameters":{"n":19,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0020","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":20}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016020,"parameters":{"n":20,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0021","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":21}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016021,"parameters":{"n":21,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0022","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":22}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016022,"parameters":{"n":22,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0023","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":23}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016023,"parameters":{"n":23,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0024","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":24}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016024,"parameters":{"n":24,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0025","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={\"n\":25}。請依 [n,coprime_residues,representative_convention] 欄序回答。","answer":{"kind":"expression","value":"(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n)","explanation_zh":"Wave5根節點核對=(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-coprime-residues","seed":202611016025,"parameters":{"n":25,"domain":"Exact_positive_integer_totient","operation":"totient-coprime-residues","semantic_template":"qf-mod32-totient-coprime-residues/{\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0026","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":1}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(1, [1], 1)","explanation_zh":"Wave5根節點核對=(1, [1], 1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016026,"parameters":{"n":1,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0027","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":2}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(2, [1], 1)","explanation_zh":"Wave5根節點核對=(2, [1], 1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016027,"parameters":{"n":2,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0028","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":3}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2], 2)","explanation_zh":"Wave5根節點核對=(3, [1, 2], 2)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016028,"parameters":{"n":3,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0029","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":4}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 3], 2)","explanation_zh":"Wave5根節點核對=(4, [1, 3], 2)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016029,"parameters":{"n":4,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0030","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":5}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(5, [1, 2, 3, 4], 4)","explanation_zh":"Wave5根節點核對=(5, [1, 2, 3, 4], 4)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016030,"parameters":{"n":5,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0031","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":6}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 5], 2)","explanation_zh":"Wave5根節點核對=(6, [1, 5], 2)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016031,"parameters":{"n":6,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0032","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":7}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(7, [1, 2, 3, 4, 5, 6], 6)","explanation_zh":"Wave5根節點核對=(7, [1, 2, 3, 4, 5, 6], 6)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016032,"parameters":{"n":7,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0033","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":8}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(8, [1, 3, 5, 7], 4)","explanation_zh":"Wave5根節點核對=(8, [1, 3, 5, 7], 4)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016033,"parameters":{"n":8,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0034","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":9}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(9, [1, 2, 4, 5, 7, 8], 6)","explanation_zh":"Wave5根節點核對=(9, [1, 2, 4, 5, 7, 8], 6)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016034,"parameters":{"n":9,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0035","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":10}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(10, [1, 3, 7, 9], 4)","explanation_zh":"Wave5根節點核對=(10, [1, 3, 7, 9], 4)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016035,"parameters":{"n":10,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0036","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":11}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10)","explanation_zh":"Wave5根節點核對=(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016036,"parameters":{"n":11,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0037","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":12}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 5, 7, 11], 4)","explanation_zh":"Wave5根節點核對=(12, [1, 5, 7, 11], 4)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016037,"parameters":{"n":12,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0038","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":13}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12)","explanation_zh":"Wave5根節點核對=(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016038,"parameters":{"n":13,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0039","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":14}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(14, [1, 3, 5, 9, 11, 13], 6)","explanation_zh":"Wave5根節點核對=(14, [1, 3, 5, 9, 11, 13], 6)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016039,"parameters":{"n":14,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0040","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":15}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(15, [1, 2, 4, 7, 8, 11, 13, 14], 8)","explanation_zh":"Wave5根節點核對=(15, [1, 2, 4, 7, 8, 11, 13, 14], 8)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016040,"parameters":{"n":15,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0041","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":16}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(16, [1, 3, 5, 7, 9, 11, 13, 15], 8)","explanation_zh":"Wave5根節點核對=(16, [1, 3, 5, 7, 9, 11, 13, 15], 8)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016041,"parameters":{"n":16,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0042","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":17}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16)","explanation_zh":"Wave5根節點核對=(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016042,"parameters":{"n":17,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0043","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":18}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(18, [1, 5, 7, 11, 13, 17], 6)","explanation_zh":"Wave5根節點核對=(18, [1, 5, 7, 11, 13, 17], 6)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016043,"parameters":{"n":18,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0044","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":19}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18)","explanation_zh":"Wave5根節點核對=(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016044,"parameters":{"n":19,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0045","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":20}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(20, [1, 3, 7, 9, 11, 13, 17, 19], 8)","explanation_zh":"Wave5根節點核對=(20, [1, 3, 7, 9, 11, 13, 17, 19], 8)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016045,"parameters":{"n":20,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0046","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":21}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12)","explanation_zh":"Wave5根節點核對=(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016046,"parameters":{"n":21,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0047","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":22}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10)","explanation_zh":"Wave5根節點核對=(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016047,"parameters":{"n":22,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0048","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":23}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22)","explanation_zh":"Wave5根節點核對=(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016048,"parameters":{"n":23,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0049","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":24}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(24, [1, 5, 7, 11, 13, 17, 19, 23], 8)","explanation_zh":"Wave5根節點核對=(24, [1, 5, 7, 11, 13, 17, 19, 23], 8)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016049,"parameters":{"n":24,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0050","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-value；資料={\"n\":25}。請依 [n,coprime_residues,phi_n] 欄序回答。","answer":{"kind":"expression","value":"(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20)","explanation_zh":"Wave5根節點核對=(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-value","seed":202611016050,"parameters":{"n":25,"domain":"Exact_positive_integer_totient","operation":"totient-value","semantic_template":"qf-mod32-totient-value/{\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0051","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":1}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, phi_1_equals_1)","explanation_zh":"Wave5根節點核對=(1, 1, phi_1_equals_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016051,"parameters":{"n":1,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0052","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":2}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(2, 1, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016052,"parameters":{"n":2,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0053","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":3}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(3, 2, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016053,"parameters":{"n":3,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0054","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":4}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(4, 2, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016054,"parameters":{"n":4,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0055","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":5}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(5, 4, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016055,"parameters":{"n":5,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0056","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":6}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(6, 2, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016056,"parameters":{"n":6,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0057","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":7}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(7, 6, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(7, 6, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016057,"parameters":{"n":7,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0058","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":8}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 4, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(8, 4, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016058,"parameters":{"n":8,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0059","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":9}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(9, 6, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(9, 6, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016059,"parameters":{"n":9,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0060","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":10}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(10, 4, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(10, 4, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016060,"parameters":{"n":10,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0061","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":11}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(11, 10, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(11, 10, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016061,"parameters":{"n":11,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0062","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":12}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(12, 4, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(12, 4, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016062,"parameters":{"n":12,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0063","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":13}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(13, 12, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(13, 12, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016063,"parameters":{"n":13,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0064","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":14}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(14, 6, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(14, 6, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016064,"parameters":{"n":14,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0065","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":15}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(15, 8, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(15, 8, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016065,"parameters":{"n":15,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0066","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":16}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(16, 8, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(16, 8, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016066,"parameters":{"n":16,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0067","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":17}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(17, 16, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(17, 16, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016067,"parameters":{"n":17,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0068","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":18}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(18, 6, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(18, 6, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016068,"parameters":{"n":18,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0069","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":19}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(19, 18, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(19, 18, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016069,"parameters":{"n":19,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0070","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":20}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(20, 8, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(20, 8, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016070,"parameters":{"n":20,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0071","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":21}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(21, 12, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(21, 12, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016071,"parameters":{"n":21,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0072","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":22}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(22, 10, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(22, 10, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016072,"parameters":{"n":22,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0073","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":23}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(23, 22, prime_phi_equals_n_minus_1)","explanation_zh":"Wave5根節點核對=(23, 22, prime_phi_equals_n_minus_1)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016073,"parameters":{"n":23,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0074","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":24}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(24, 8, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(24, 8, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016074,"parameters":{"n":24,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0075","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-boundary-prime；資料={\"n\":25}。請依 [n,phi_n,classification] 欄序回答。","answer":{"kind":"expression","value":"(25, 20, composite_no_prime_shortcut)","explanation_zh":"Wave5根節點核對=(25, 20, composite_no_prime_shortcut)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-boundary-prime","seed":202611016075,"parameters":{"n":25,"domain":"Exact_positive_integer_totient","operation":"totient-boundary-prime","semantic_template":"qf-mod32-totient-boundary-prime/{\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0076","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":3}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 1, 2, 2, 2, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 3, 1, 2, 2, 2, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016076,"parameters":{"m":2,"n":3,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0077","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":5}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 1, 4, 4, 4, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 5, 1, 4, 4, 4, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016077,"parameters":{"m":2,"n":5,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0078","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":7}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 7, 1, 6, 6, 6, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 7, 1, 6, 6, 6, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016078,"parameters":{"m":2,"n":7,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0079","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":9}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 9, 1, 6, 6, 6, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 9, 1, 6, 6, 6, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016079,"parameters":{"m":2,"n":9,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0080","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":11}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 11, 1, 10, 10, 10, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 11, 1, 10, 10, 10, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016080,"parameters":{"m":2,"n":11,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0081","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":13}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 13, 1, 12, 12, 12, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 13, 1, 12, 12, 12, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016081,"parameters":{"m":2,"n":13,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0082","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":15}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 15, 1, 8, 8, 8, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 15, 1, 8, 8, 8, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016082,"parameters":{"m":2,"n":15,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0083","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":17}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 17, 1, 16, 16, 16, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 17, 1, 16, 16, 16, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016083,"parameters":{"m":2,"n":17,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0084","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":19}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 19, 1, 18, 18, 18, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 19, 1, 18, 18, 18, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016084,"parameters":{"m":2,"n":19,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0085","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":21}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 21, 1, 12, 12, 12, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 21, 1, 12, 12, 12, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016085,"parameters":{"m":2,"n":21,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0086","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":23}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 23, 1, 22, 22, 22, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 23, 1, 22, 22, 22, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016086,"parameters":{"m":2,"n":23,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0087","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":25}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 25, 1, 20, 20, 20, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 25, 1, 20, 20, 20, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016087,"parameters":{"m":2,"n":25,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0088","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":27}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 27, 1, 18, 18, 18, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 27, 1, 18, 18, 18, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016088,"parameters":{"m":2,"n":27,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0089","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":29}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 29, 1, 28, 28, 28, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 29, 1, 28, 28, 28, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016089,"parameters":{"m":2,"n":29,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0090","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":31}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 31, 1, 30, 30, 30, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 31, 1, 30, 30, 30, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016090,"parameters":{"m":2,"n":31,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0091","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":33}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 33, 1, 20, 20, 20, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 33, 1, 20, 20, 20, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016091,"parameters":{"m":2,"n":33,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0092","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":35}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 35, 1, 24, 24, 24, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 35, 1, 24, 24, 24, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016092,"parameters":{"m":2,"n":35,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0093","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":37}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 37, 1, 36, 36, 36, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 37, 1, 36, 36, 36, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016093,"parameters":{"m":2,"n":37,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0094","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":39}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 39, 1, 24, 24, 24, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 39, 1, 24, 24, 24, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016094,"parameters":{"m":2,"n":39,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0095","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":41}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 41, 1, 40, 40, 40, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 41, 1, 40, 40, 40, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016095,"parameters":{"m":2,"n":41,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0096","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":43}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 43, 1, 42, 42, 42, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 43, 1, 42, 42, 42, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016096,"parameters":{"m":2,"n":43,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0097","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":45}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 45, 1, 24, 24, 24, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 45, 1, 24, 24, 24, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016097,"parameters":{"m":2,"n":45,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0098","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":47}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 47, 1, 46, 46, 46, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 47, 1, 46, 46, 46, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016098,"parameters":{"m":2,"n":47,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0099","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":49}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 49, 1, 42, 42, 42, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(2, 49, 1, 42, 42, 42, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016099,"parameters":{"m":2,"n":49,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r06-euler-totient-function-0100","batch_id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-totient-function"],"title_zh":"Euler φ函數","stem_zh":"正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":3,\"n\":4}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 2, 2, 4, 4, 是, instance_only_with_coprimality)","explanation_zh":"Wave5根節點核對=(3, 4, 2, 2, 4, 4, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler φ函數"]},"variant":{"family_id":"qf-mod32-totient-multiplicativity-instance","seed":202611016100,"parameters":{"m":3,"n":4,"domain":"Exact_positive_integer_totient","operation":"totient-multiplicativity-instance","semantic_template":"qf-mod32-totient-multiplicativity-instance/{\"m\":3,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}