{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0042", "pilot_number": "R0041-21317", "number": 21317, "family_id": "qf-mod39-upf-canonical-factorization", "title": { "zh-Hant": "第 21317 題：整數唯一質因數分解", "en": "Question 21317: Complete Trial Division Produces the Canonical Prime Multiset" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0042", "batch_id": "batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-unique-prime-factorization" ], "title_zh": "整數唯一質因數分解", "stem_zh": "完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":354}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, 是, complete_trial_division_range)", "explanation_zh": "Wave6第二層核對=(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "整數唯一質因數分解" ] }, "variant": { "family_id": "qf-mod39-upf-canonical-factorization", "seed": 202611086042, "parameters": { "n": 354, "domain": "Exact_integer_unique_prime_factorization", "operation": "upf-canonical-factorization", "semantic_template": "qf-mod39-upf-canonical-factorization/{\"n\":354}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0042", "batch_id": "batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "2faaea4aeba80a93882e9e71864968c1d801f0c56ac3a691e70ef274ddfdf3a8", "family_id": "qf-mod39-upf-canonical-factorization", "seed": 202611086042, "semantic_template": "qf-mod39-upf-canonical-factorization/{\"n\":354}" }, "fields": { "title": "Complete Trial Division Produces the Canonical Prime Multiset", "stem": "Use positive integer n=354>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).", "answer": { "kind": "expression", "value": "(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)", "explanation": "Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)." }, "skills": [ "Complete Trial Division Produces the Canonical Prime Multiset" ], "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": "完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":354}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use positive integer n=354>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=upf-canonical-factorization;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[n, prime_multiset, exponents, product, complete_factorization_certificate, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記保留；此處直接列有序輸出欄位，不依賴JavaScript。欄位或tag不是額外數學證據，固定No保留原聲稱範圍。", "en": "Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope." }, "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": "ORIGINAL positive integer absolute magnitude=354>1", "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted", "relation": "implies", "rule_id": "upf_factorization_remaining_product_invariant", "conditions": [], "explanation": { "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。", "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime." }, "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": "upf extract divisor2 occurrence0 := div(354, 2)", "after": "upf extract divisor2 occurrence0 = 177", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。", "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "354", "2" ], "output": "177", "label": "upf extract divisor2 occurrence0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "next divisor=2;previous remaining=354;all smaller divisors exhausted", "after": "extracted multiplicity=1;next remaining=177;remaining modulo2=1;extracted multiset=[2]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "upf extract divisor3 occurrence1 := div(177, 3)", "after": "upf extract divisor3 occurrence1 = 59", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。", "en": "Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "177", "3" ], "output": "59", "label": "upf extract divisor3 occurrence1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "next divisor=3;previous remaining=177;all smaller divisors exhausted", "after": "extracted multiplicity=1;next remaining=59;remaining modulo3=2;extracted multiset=[2, 3]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "next divisor=4;previous remaining=59;all smaller divisors exhausted", "after": "extracted multiplicity=0;next remaining=59;remaining modulo4=3;extracted multiset=[2, 3]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "next divisor=5;previous remaining=59;all smaller divisors exhausted", "after": "extracted multiplicity=0;next remaining=59;remaining modulo5=4;extracted multiset=[2, 3]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "next divisor=6;previous remaining=59;all smaller divisors exhausted", "after": "extracted multiplicity=0;next remaining=59;remaining modulo6=5;extracted multiset=[2, 3]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "next divisor=7;previous remaining=59;all smaller divisors exhausted", "after": "extracted multiplicity=0;next remaining=59;remaining modulo7=3;extracted multiset=[2, 3]", "relation": "implies", "rule_id": "upf_complete_trial_division_remaining_invariant", "conditions": [], "explanation": { "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。", "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "upf canonical complete product factor0 := mul(1, 2)", "after": "upf canonical complete product factor0 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。", "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "upf canonical complete product factor0" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "upf canonical complete product factor1 := mul(2, 3)", "after": "upf canonical complete product factor1 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。", "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "upf canonical complete product factor1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "upf canonical complete product factor2 := mul(6, 59)", "after": "upf canonical complete product factor2 = 354", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。", "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "59" ], "output": "354", "label": "upf canonical complete product factor2" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "exhausted trial divisors below8;terminal remaining=59;d*d>remaining;ORIGINAL n=354", "after": "complete increasing prime multiset=[2, 3, 59];full product=354;a composite terminal remainder would have an already tested smaller divisor", "relation": "implies", "rule_id": "upf_final_prime_remainder_complete_existence_instance", "conditions": [], "explanation": { "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。", "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors." }, "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": "S015", "uses": [ "G001", "S014" ], "before": "ORIGINAL n=354;complete prime multiset=[2, 3, 59]", "after": "ALL [prime,exponent] rows=[[2, 1], [3, 1], [59, 1]];product=354;no multiplicities discarded", "relation": "implies", "rule_id": "upf_exponents_keep_every_prime_multiplicity", "conditions": [], "explanation": { "zh-Hant": "逐不同prime保完整重數，不把multiset改成set。後續唯一性比較必須比較每個exponent，不只不同prime種類。", "en": "Retain every multiplicity,not just distinct prime support.Uniqueness compares all exponents,not only which prime species occur." }, "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": [ 354, [ 2, 3, 59 ], [ [ 2, 1 ], [ 3, 1 ], [ 59, 1 ] ], 354, true, "complete_trial_division_range" ] }, "conclusion": { "canonical_value": "(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, 是, complete_trial_division_range)", "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-20261014-advance-20261108-r06-unique-prime-factorization-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/questions.jsonl", "batch_sha256": "51fa926e07cc6c20069db4e8080bcf373df4bf3de100f279fae9cd21c30f207d", "original_line": 42, "original_line_sha256": "23f4121a465774944fe86d26834aec5586ad8e2b5a0bed860caadf72f16288fd", "question_compact_sha256": "2faaea4aeba80a93882e9e71864968c1d801f0c56ac3a691e70ef274ddfdf3a8", "english_row_compact_sha256": "e83147193a607594a6444f358b9f5f37e4473672d6916a374a2aa93c5f9e3891", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations.en.jsonl" }, "mko_bindings": { "mko-unique-prime-factorization": { "sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff", "english_sha256": "daa563be289db97eeabc811b523c607b1c88cb0a7c4b7b320a62dee46bcc27ac", "original_path": "public/data/mko/source13-20261018-unique-prime-factorization.json", "english_path": "public/data/i18n/editions/mko-unique-prime-factorization.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "2bda8f45e99ed8658b697be77ed3e5ec90ceff1b4f72fa7aa6a47d1a0c6254cf", "original_candidate_sha256": "2bc44a1d53c7cf6635a371e1073e9124060722143b0ec01ace5f12b064ce223d", "original_candidate_line_sha256": "2bc44a1d53c7cf6635a371e1073e9124060722143b0ec01ace5f12b064ce223d", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-unique-prime-factorization": "3efccc30de9556546dd853570ad8776c24689cd9507cd911b9139b0b825e9b76" } } }, "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-20261026-009-p42", "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": "8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a", "question_first_witness_sha256": "ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 15, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2af12fc540230ded2887426432857f66d4c99a62bf2a43d2ea0d93da2dfa68fd", "arithmetic_steps": 5 }, "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." } }