{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r04-prime-number-0009",
  "pilot_number": "R0017-09109",
  "number": 9109,
  "family_id": "qf-ext18-prime-divisor-classification",
  "title": {
    "zh-Hant": "第 9109 題：自然數質數與因數",
    "en": "Question 9109: Complete Positive Divisors and Prime Qualification"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r04-prime-number-0009",
    "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-divisor-classification；資料={\"p\":8}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(8, [1, 2, 4, 8], 是, 否)",
      "explanation_zh": "擴展基礎核對=(8, [1, 2, 4, 8], 是, 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": 2,
      "skills": [
        "自然數質數與因數"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-prime-divisor-classification",
      "seed": 202610184009,
      "parameters": {
        "p": 8,
        "domain": "Nat_prime",
        "operation": "prime-divisor-classification",
        "semantic_template": "qf-ext18-prime-divisor-classification/{\"p\":8}"
      }
    },
    "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-0009",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dad5b790fdb79bbb27d3631bcd65a890b17219b1407c8380c1a6d600971d296d",
      "family_id": "qf-ext18-prime-divisor-classification",
      "seed": 202610184009,
      "semantic_template": "qf-ext18-prime-divisor-classification/{\"p\":8}"
    },
    "fields": {
      "title": "Complete Positive Divisors and Prime Qualification",
      "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. p=8. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).",
      "answer": {
        "kind": "expression",
        "value": "(8, [1, 2, 4, 8], Yes, No)",
        "explanation": "Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [1, 2, 4, 8], Yes, No)."
      },
      "skills": [
        "Complete Positive Divisors and Prime Qualification"
      ],
      "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-divisor-classification；資料={\"p\":8}。請依 [p,positive_divisors_or_all,p_at_least_2,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. p=8. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?)."
      }
    }
  ],
  "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": "positive factor pair candidate1:natural dividend=8;positive divisor=1",
      "after": "8=8·1+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "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": {
        "operation": "euclidean_division",
        "inputs": [
          "8",
          "1",
          "8"
        ],
        "output": "0",
        "label": "positive factor pair candidate1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "candidate d=1;quotient=8;remainder=0",
      "after": "factor pair=[1, 8]",
      "relation": "implies",
      "rule_id": "complete_positive_factor_pair_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零餘數才把d與n/d同時加入；平方根重合時去重。任一因數對至少一端≤sqrt(n)，故配對覆蓋所有正因數。",
        "en": "Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "positive factor pair candidate2:natural dividend=8;positive divisor=2",
      "after": "8=4·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "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": {
        "operation": "euclidean_division",
        "inputs": [
          "8",
          "2",
          "4"
        ],
        "output": "0",
        "label": "positive factor pair candidate2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate d=2;quotient=4;remainder=0",
      "after": "factor pair=[2, 4]",
      "relation": "implies",
      "rule_id": "complete_positive_factor_pair_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零餘數才把d與n/d同時加入；平方根重合時去重。任一因數對至少一端≤sqrt(n)，故配對覆蓋所有正因數。",
        "en": "Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "all positive divisors=[1, 2, 4, 8];n=8",
      "after": "n>=2=是;exactly1 and n=否;prime=否",
      "relation": "implies",
      "rule_id": "prime_positive_divisors_with_lower_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整正因數恰為1與n且n≥2才是質數；n=1的集合只有一個1，不把重複列出的[1,1]當兩個因數。",
        "en": "Primality requires n≥2 and the complete positive-divisor set exactly{1,n}.At n=1 there is only one divisor1,not two copies in[1,1]."
      },
      "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": [
      8,
      [
        1,
        2,
        4,
        8
      ],
      true,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(8, [1, 2, 4, 8], 是, 否)",
    "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-r04-prime-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r04-prime-number-foundations/questions.jsonl",
      "batch_sha256": "39b1f12c357ca8c69d6a9660a72667958bd462bae4faf3ab07254018c9cfafe4",
      "original_line": 9,
      "original_line_sha256": "87bd5cba16364f811556de5107c262e806f76025ff123b2d7c17e1209815ee8f",
      "question_compact_sha256": "dad5b790fdb79bbb27d3631bcd65a890b17219b1407c8380c1a6d600971d296d",
      "english_row_compact_sha256": "b18c5a2733317104d285ba201906b0ed55ceb4c9a8c5ebf21a9081d22bcd0f3a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r04-prime-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r04-prime-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-prime-number": {
        "sha256": "5ebeb1330ba52c9685111730d0c0c2255e653887f8869f119ebbbb767ed139b5",
        "english_sha256": "9c7980d6fc0591adf0ccc93fefbab485b5b58c30d23d24e43095b2a8f18fd5aa",
        "original_path": "public/data/mko/prime-number.json",
        "english_path": "public/data/i18n/editions/mko-prime-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20e59f30c0b6218e6d9614c16170d770e56a2f1e4ae5ce39d0a8a3db206afa61",
    "original_candidate_sha256": "6e7abcea11bb7d966c3dd1fcaece6cc00f98c57862ed173eae5dd822abcef140",
    "original_candidate_line_sha256": "6e7abcea11bb7d966c3dd1fcaece6cc00f98c57862ed173eae5dd822abcef140"
  },
  "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-20261026-006-p18",
    "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": "abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8",
      "question_first_witness_sha256": "43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2a6f2b527c758311a57bc3d48da6b794f6624ac35a220f8f248efeb5d68feab9",
      "arithmetic_steps": 2
    },
    "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."
  }
}