{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261018-r02-natural-divisibility-0096", "pilot_number": "R0016-08846", "number": 8846, "family_id": "qf-ext18-divisibility-transitivity", "title": { "zh-Hant": "第 8846 題：自然數整除與零邊界", "en": "Question 8846: Compose Two Given Natural Divisibility Witnesses" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261018-r02-natural-divisibility-0096", "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-natural-divisibility" ], "title_zh": "自然數整除與零邊界", "stem_zh": "自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":0,\"l\":4}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。", "answer": { "kind": "expression", "value": "(2, 0, 0, 0, 4, 0, 是)", "explanation_zh": "擴展基礎核對=(2, 0, 0, 0, 4, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。" }, "difficulty": { "band": "upper_secondary", "level": 3, "skills": [ "自然數整除與零邊界" ] }, "variant": { "family_id": "qf-ext18-divisibility-transitivity", "seed": 202610182096, "parameters": { "a": 2, "k": 0, "l": 4, "domain": "Nat_divisibility", "operation": "divisibility-transitivity", "semantic_template": "qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":0,\"l\":4}" } }, "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-r02-natural-divisibility-0096", "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1da0c82ada31da5a9cd8ff0f78f02ac552fd689489bf726238fc5d04f6c9bfa5", "family_id": "qf-ext18-divisibility-transitivity", "seed": 202610182096, "semantic_template": "qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":0,\"l\":4}" }, "fields": { "title": "Compose Two Given Natural Divisibility Witnesses", "stem": "Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=0, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).", "answer": { "kind": "expression", "value": "(2, 0, 0, 0, 4, 0, Yes)", "explanation": "Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 0, 0, 0, 4, 0, Yes)." }, "skills": [ "Compose Two Given Natural Divisibility Witnesses" ], "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": "自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":0,\"l\":4}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。", "en": "Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=0, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "natural numbers include0;b is divisible by a iff b=a*k for some natural k", "after": "0 divides only0;every natural divides0", "relation": "implies", "rule_id": "natural_divisibility_zero_scope", "conditions": [], "explanation": { "zh-Hant": "整除是存在乘數的關係，不先假定除數正或非零；a∣0包含a=0。正因數清單另排除0，兩個問題不混用。", "en": "Divisibility is an existential multiplier relation without presuming a positive or nonzero divisor.Every natural,including0,divides0.A positive-divisor inventory separately excludes0." }, "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": "first multiple b=a*k := mul(2, 0)", "after": "first multiple b=a*k = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。", "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 first multiple b=a*k，不以小數近似替代。", "en": "Recompute first multiple b=a*k with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "0" ], "output": "0", "label": "first multiple b=a*k" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "second multiple c=b*l := mul(0, 4)", "after": "second multiple c=b*l = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。", "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 second multiple c=b*l，不以小數近似替代。", "en": "Recompute second multiple c=b*l with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "4" ], "output": "0", "label": "second multiple c=b*l" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "composed multiplier k*l := mul(0, 4)", "after": "composed multiplier k*l = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。", "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 composed multiplier k*l，不以小數近似替代。", "en": "Recompute composed multiplier k*l with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "4" ], "output": "0", "label": "composed multiplier k*l" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "a times composed multiplier := mul(2, 0)", "after": "a times composed multiplier = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。", "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 a times composed multiplier，不以小數近似替代。", "en": "Recompute a times composed multiplier with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "0" ], "output": "0", "label": "a times composed multiplier" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "composed divisibility:divide natural 0 by positive 2", "after": "0=0·2+0", "relation": "implies", "rule_id": "natural_euclidean_division_witness", "conditions": [], "explanation": { "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r