{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0022", "pilot_number": "R0066-33822", "number": 33822, "family_id": "qf-mod47-morder-qualification", "title": { "zh-Hant": "第 33822 題：模n乘法階", "en": "Question 33822: Normalize and Qualify a Unit before Defining Multiplicative Order" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0022", "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-multiplicative-order-mod-n" ], "title_zh": "模n乘法階", "stem_zh": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":24,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(24, 5, 5, 1, 8, 是, unit_mod_n_qualified)", "explanation_zh": "Wave6第十層核對=(24, 5, 5, 1, 8, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "模n乘法階" ] }, "variant": { "family_id": "qf-mod47-morder-qualification", "seed": 202611167022, "parameters": { "n": 24, "a": 5, "domain": "Exact_multiplicative_order_mod_n", "operation": "morder-qualification", "semantic_template": "qf-mod47-morder-qualification/{\"a\":[5,1],\"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": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0022", "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9877fc10baaa792c759fc4e8d7ef0f4f86e6353294e716fdf7c974ef19e11787", "family_id": "qf-mod47-morder-qualification", "seed": 202611167022, "semantic_template": "qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":24}" }, "fields": { "title": "Normalize and Qualify a Unit before Defining Multiplicative Order", "stem": "Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).", "answer": { "kind": "expression", "value": "(24, 5, 5, 1, 8, Yes, unit_mod_n_qualified)", "explanation": "Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (24, 5, 5, 1, 8, Yes, unit_mod_n_qualified)." }, "skills": [ "Normalize and Qualify a Unit before Defining Multiplicative Order" ], "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": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":24,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod47-morder-qualification, [a, domain, n, operation]]", "after": "derived = [morder-qualification, Exact_multiplicative_order_mod_n]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [ORIGINAL representative, 5, 24]", "after": "derived = [0, 5]", "relation": "implies", "rule_id": "Finite_MOD_euclidean_quotient_remainder", "conditions": [], "explanation": { "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder