{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r03-euclidean-algorithm-0078",
  "pilot_number": "R0036-18878",
  "number": 18878,
  "family_id": "qf-mod36-euclid-zero-input-boundary",
  "title": {
    "zh-Hant": "第 18878 題：Euclidean演算法",
    "en": "Question 18878: Zero Inputs Have an Explicit Gcd Convention without Division by Zero"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r03-euclidean-algorithm-0078",
    "batch_id": "batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclidean-algorithm"
    ],
    "title_zh": "Euclidean演算法",
    "stem_zh": "exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)",
      "explanation_zh": "Wave5第五層核對=(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Euclidean演算法"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-euclid-zero-input-boundary",
      "seed": 202611053078,
      "parameters": {
        "a": -3,
        "b": 0,
        "domain": "Exact_integer_Euclidean_chain",
        "operation": "euclid-zero-input-boundary",
        "semantic_template": "qf-mod36-euclid-zero-input-boundary/{\"a\":-3,\"b\":0}"
      }
    },
    "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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r03-euclidean-algorithm-0078",
    "batch_id": "batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d7001567d6efc2d749582be61bb5345f9e74273fd346e17915bfae267c04094d",
      "family_id": "qf-mod36-euclid-zero-input-boundary",
      "seed": 202611053078,
      "semantic_template": "qf-mod36-euclid-zero-input-boundary/{\"a\":-3,\"b\":0}"
    },
    "fields": {
      "title": "Zero Inputs Have an Explicit Gcd Convention without Division by Zero",
      "stem": "Use original integers a=-3, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).",
      "answer": {
        "kind": "expression",
        "value": "(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)",
        "explanation": "Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)."
      },
      "skills": [
        "Zero Inputs Have an Explicit Gcd Convention without Division by Zero"
      ],
      "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": "exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use original integers a=-3, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=euclid-zero-input-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[a, b, gcd, x, y, linear_combination, convention]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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 signed inputs a=-3,b=0",
      "after": "initial nonnegative chain values=[3, 0];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input",
      "relation": "implies",
      "rule_id": "euclid_signed_inputs_absolute_chain_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "公因數的整除不受符號改變影響；gcd取非負。一般chain/invariant要求b非零，零邊界另保留(0,0)約定，不擅自套該限制。",
        "en": "Divisibility is unaffected by input signs,and the gcd is nonnegative.Ordinary chain/invariant families require b≠0;the zero boundary separately retains its(0,0)convention."
      },
      "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": "complete division rows=[];terminal pair=[3, 0]",
      "after": "gcd(a,b)=gcd(3,0)=3;all divisions used positive divisors;nonnegative gcd convention",
      "relation": "implies",
      "rule_id": "euclid_terminal_gcd_not_extra_zero_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "終止時所有共同因數等於terminal left的因數，非負最大公因數就是left；(0,0)的0另由明示約定給出。不新增除以零的虛假一列。",
        "en": "At termination common divisors are those of the terminal left value,giving its nonnegative gcd.The(0,0)value0 is an explicit separate convention.Do not append a fictitious division by zero."
      },
      "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": "Euclid restore original a sign in coefficient := mul(-1, 1)",
      "after": "Euclid restore original a sign in coefficient = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid restore original a sign in coefficient，不以小數近似替代。",
          "en": "Recompute Euclid restore original a sign in coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "Euclid restore original a sign in coefficient"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Euclid restore original b sign in coefficient := mul(1, 0)",
      "after": "Euclid restore original b sign in coefficient = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid restore original b sign in coefficient，不以小數近似替代。",
          "en": "Recompute Euclid restore original b sign in coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Euclid restore original b sign in coefficient"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Euclid ORIGINAL signed a times x := mul(-3, -1)",
      "after": "Euclid ORIGINAL signed a times x = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid ORIGINAL signed a times x，不以小數近似替代。",
          "en": "Recompute Euclid ORIGINAL signed a times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "-1"
        ],
        "output": "3",
        "label": "Euclid ORIGINAL signed a times x"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Euclid ORIGINAL signed b times y := mul(0, 0)",
      "after": "Euclid ORIGINAL signed b times y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid ORIGINAL signed b times y，不以小數近似替代。",
          "en": "Recompute Euclid ORIGINAL signed b times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Euclid ORIGINAL signed b times y"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Euclid final signed Bezout linear combination := sum(3, 0)",
      "after": "Euclid final signed Bezout linear combination = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid final signed Bezout linear combination，不以小數近似替代。",
          "en": "Recompute Euclid final signed Bezout linear combination with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "Euclid final signed Bezout linear combination"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original a=-3,b=0;deterministic absolute-chain coefficients=[1, 0]",
      "after": "sign-restored source coefficients x=-1,y=0;a*x+b*y=3=nonnegative gcd",
      "relation": "implies",
      "rule_id": "euclid_deterministic_signed_bezout_not_just_any_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "來源要求這條extended chain的確定係數，雖然其他Bézout係數也可能正確，仍不能替換來源答案。零符號保+1；(0,0)給原係數(1,0)及0=0，不聲稱唯一。",
        "en": "Retain the deterministic coefficients of this source extended chain even though other Bezout pairs may be valid.Use sign+1 for zero;(0,0)keeps(1,0)and0=0,without a uniqueness claim."
      },
      "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": [
      -3,
      0,
      3,
      -1,
      0,
      3,
      "no_division_by_zero_executed"
    ]
  },
  "conclusion": {
    "canonical_value": "(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)",
    "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."
    }
  },
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    "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."
  },
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      "formalization": "not_formalized",
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      "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."
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}