{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261018-r02-natural-divisibility-0026", "pilot_number": "R0016-08801", "number": 8801, "family_id": "qf-ext18-divisibility-divisor-set", "title": { "zh-Hant": "第 8801 題：自然數整除與零邊界", "en": "Question 8801: Complete Positive Divisor Inventory Including the Zero Case" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261018-r02-natural-divisibility-0026", "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-divisor-set；資料={\"b\":0}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。", "answer": { "kind": "expression", "value": "(0, 所有正自然數, 無限)", "explanation_zh": "擴展基礎核對=(0, 所有正自然數, 無限)；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": 1, "skills": [ "自然數整除與零邊界" ] }, "variant": { "family_id": "qf-ext18-divisibility-divisor-set", "seed": 202610182026, "parameters": { "b": 0, "domain": "Nat_divisibility", "operation": "divisibility-divisor-set", "semantic_template": "qf-ext18-divisibility-divisor-set/{\"b\":0}" } }, "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-0026", "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7ac5eb8a1d773434df82883bc195b25e920be10716de9865c2f9488e68b65ea0", "family_id": "qf-ext18-divisibility-divisor-set", "seed": 202610182026, "semantic_template": "qf-ext18-divisibility-divisor-set/{\"b\":0}" }, "fields": { "title": "Complete Positive Divisor Inventory Including the Zero Case", "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. b=0. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.", "answer": { "kind": "expression", "value": "(0, all-positive-naturals, infinite)", "explanation": "For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, all-positive-naturals, infinite)." }, "skills": [ "Complete Positive Divisor Inventory Including the Zero Case" ], "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-divisor-set；資料={\"b\":0}。請依 [b,positive_divisors_or_all,count_or_無限] 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. b=0. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0." } } ], "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": "b=0;positive candidate d is arbitrary", "after": "d*0=0 for every positive d;positive divisors=所有正自然數;count=無限", "relation": "implies", "rule_id": "zero_all_positive_divisors_unbounded", "conditions": [], "explanation": { "zh-Hant": "任意正d都有乘數0，故正因數集合是所有正自然數且無限；不是只列一個有限測試區間，也不把除數0混入正因數。", "en": "Every positive d has multiplier0,so the positive-divisor set is all positive naturals and is infinite.This is not a finite-window sample,and divisor0 is not a positive divisor." }, "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 } ], "math_model": { "result": [ 0, "所有正自然數", "無限" ] }, "conclusion": { "canonical_value": "(0, 所有正自然數, 無限)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/questions.jsonl", "batch_sha256": "85abf07ffd0c46a8235728cd1ab694010d535a2683ea0040042f9de7a1360357", "original_line": 26, "original_line_sha256": "e047c82fa369bf523c63024e878f54e38a6453d7410ba117df5aeadda4fdd58a", "question_compact_sha256": "7ac5eb8a1d773434df82883bc195b25e920be10716de9865c2f9488e68b65ea0", "english_row_compact_sha256": "df0691e14d948adf219af5511672206198835cc3b15f132d0ddb9261a7f5b4b4", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r02-natural-divisibility-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r02-natural-divisibility-foundations.en.jsonl" }, "mko_bindings": { "mko-natural-divisibility": { "sha256": "2e5d00faa3ae3031b2cb2a650b12db0ab8e8aa1b59a7ea6b2350f49a792758d7", "english_sha256": "acae542d73cc84e0fd3fd3443b35f99bca7758dce17f8b8ab65ee9957193e395", "original_path": "public/data/mko/natural-divisibility.json", "english_path": "public/data/i18n/editions/mko-natural-divisibility.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "8e8954cf690c44da1ec1efa08994a485c73ca8706b7865d854e3f2cdfd4ca89f", "original_candidate_sha256": "0fd0fa9a609007684a5f71bd073e6bc84c3074583876ba3798324cdc11c4a671", "original_candidate_line_sha256": "0fd0fa9a609007684a5f71bd073e6bc84c3074583876ba3798324cdc11c4a671" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-005-p17", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2", "question_first_witness_sha256": "198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c7a6dc87673cb7b60df90971cf29ce24d619a5264b42a935a2c708e6832e062a", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }