{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261018-r04-prime-number-0029", "pilot_number": "R0017-09179", "number": 9179, "family_id": "qf-ext18-prime-smallest-factor", "title": { "zh-Hant": "第 9179 題：自然數質數與因數", "en": "Question 9179: Least Divisor at Least Two and Primality" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261018-r04-prime-number-0029", "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-prime-number" ], "title_zh": "自然數質數與因數", "stem_zh": "自然數質數。操作=prime-smallest-factor；資料={\"n\":33}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。", "answer": { "kind": "expression", "value": "(33, 3, 否)", "explanation_zh": "擴展基礎核對=(33, 3, 否)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0與1不是質數；因數檢查必須完整到平方根界。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "自然數質數與因數" ] }, "variant": { "family_id": "qf-ext18-prime-smallest-factor", "seed": 202610184029, "parameters": { "n": 33, "domain": "Nat_prime", "operation": "prime-smallest-factor", "semantic_template": "qf-ext18-prime-smallest-factor/{\"n\":33}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_arithmetic", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261018-r04-prime-number-0029", "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6a2776204241784bce9f2e4e9c4188f96e383f0a72da2c8477287df26e896c29", "family_id": "qf-ext18-prime-smallest-factor", "seed": 202610184029, "semantic_template": "qf-ext18-prime-smallest-factor/{\"n\":33}" }, "fields": { "title": "Least Divisor at Least Two and Primality", "stem": "Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=33. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.", "answer": { "kind": "expression", "value": "(33, 3, No)", "explanation": "Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, 3, No)." }, "skills": [ "Least Divisor at Least Two and Primality" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "自然數質數。操作=prime-smallest-factor；資料={\"n\":33}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。", "en": "Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=33. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "natural-number prime protocol;0 and1 are not prime", "after": "a failed candidate witness alone is not a primality proof", "relation": "implies", "rule_id": "prime_complete_evidence_scope", "conditions": [], "explanation": { "zh-Hant": "自然數含0；質數判準須保留n≥2與完整因數界。單一因數候選失敗只否定該見證，不能直接斷言質數。", "en": "Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "phase=given n;n=33", "after": "n>=2=是", "relation": "implies", "rule_id": "prime_lower_bound_guard", "conditions": [], "explanation": { "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。", "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "given n trial divisor2:natural dividend=33;positive divisor=2", "after": "33=16·2+1", "relation": "implies", "rule_id": "positive_euclidean_division", "conditions": [], "explanation": { "zh-Hant": "正除數下完整列商餘等式，0≤r