第 18813 題：Euclidean演算法

原題與已知條件

exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={"a":677,"b":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={"a":677,"b":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-euclidean-algorithm

逐步推導

source operation=euclid-bezout-certificate;original snapshot and mathematics retained

explicit ordered source output fields=[gcd, x, y, linear_combination, equal, certificate]

original signed inputs a=677,b=113

initial nonnegative chain values=[677, 113];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(677, 113)

Euclid step0 exact ratio = 677/113

Euclid step0 quotient times divisor := mul(5, 113)

Euclid step0 quotient times divisor = 565

Euclid step0 remainder := sub(677, 565)

Euclid step0 remainder = 112

step=0;left=677;positive right=113;exact ratio=677/113;floor quotient=5

left=right*q+r with r=112 in[0,113);gcd pair=[1, 1]

Euclid step0 first coefficient multiple := mul(5, 0)

Euclid step0 first coefficient multiple = 0

Euclid step0 first coefficient difference := sub(1, 0)

Euclid step0 first coefficient difference = 1

Euclid step0 second coefficient multiple := mul(5, 1)

Euclid step0 second coefficient multiple = 5

Euclid step0 second coefficient difference := sub(0, 5)

Euclid step0 second coefficient difference = -5

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=5

next remainder coefficient pair=[1, -5];abs(a)*s_next+abs(b)*t_next=112

Euclid step1 exact ratio := div(113, 112)

Euclid step1 exact ratio = 113/112

Euclid step1 quotient times divisor := mul(1, 112)

Euclid step1 quotient times divisor = 112

Euclid step1 remainder := sub(113, 112)

Euclid step1 remainder = 1

step=1;left=113;positive right=112;exact ratio=113/112;floor quotient=1

left=right*q+r with r=1 in[0,112);gcd pair=[1, 1]

Euclid step1 first coefficient multiple := mul(1, 1)

Euclid step1 first coefficient multiple = 1

Euclid step1 first coefficient difference := sub(0, 1)

Euclid step1 first coefficient difference = -1

Euclid step1 second coefficient multiple := mul(1, -5)

Euclid step1 second coefficient multiple = -5

Euclid step1 second coefficient difference := sub(1, -5)

Euclid step1 second coefficient difference = 6

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -5]];same quotient=1

next remainder coefficient pair=[-1, 6];abs(a)*s_next+abs(b)*t_next=1

Euclid step2 exact ratio := div(112, 1)

Euclid step2 exact ratio = 112

Euclid step2 quotient times divisor := mul(112, 1)

Euclid step2 quotient times divisor = 112

Euclid step2 remainder := sub(112, 112)

Euclid step2 remainder = 0

step=2;left=112;positive right=1;exact ratio=112;floor quotient=112

left=right*q+r with r=0 in[0,1);gcd pair=[1, 1]

Euclid step2 first coefficient multiple := mul(112, -1)

Euclid step2 first coefficient multiple = -112

Euclid step2 first coefficient difference := sub(1, -112)

Euclid step2 first coefficient difference = 113

Euclid step2 second coefficient multiple := mul(112, 6)

Euclid step2 second coefficient multiple = 672

Euclid step2 second coefficient difference := sub(-5, 672)

Euclid step2 second coefficient difference = -677

step=2;two original-absolute-input coefficient pairs=[[1, -5], [-1, 6]];same quotient=112

next remainder coefficient pair=[113, -677];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[677, 113, 5, 112], [113, 112, 1, 1], [112, 1, 112, 0]];terminal pair=[1, 0]

gcd(a,b)=gcd(1,0)=1;all divisions used positive divisors;nonnegative gcd convention

Euclid restore original a sign in coefficient := mul(1, -1)

Euclid restore original a sign in coefficient = -1

Euclid restore original b sign in coefficient := mul(1, 6)

Euclid restore original b sign in coefficient = 6

Euclid ORIGINAL signed a times x := mul(677, -1)

Euclid ORIGINAL signed a times x = -677

Euclid ORIGINAL signed b times y := mul(113, 6)

Euclid ORIGINAL signed b times y = 678

Euclid final signed Bezout linear combination := sum(-677, 678)

Euclid final signed Bezout linear combination = 1

original a=677,b=113;deterministic absolute-chain coefficients=[-1, 6]

sign-restored source coefficients x=-1,y=6;a*x+b*y=1=nonnegative gcd

逐步反向核對

• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 以精確有理數重算 Euclid final signed Bezout linear combination，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euclid ORIGINAL signed b times y，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Euclid ORIGINAL signed a times x，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Euclid restore original b sign in coefficient，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Euclid restore original a sign in coefficient，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 Euclid step2 second coefficient difference，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euclid step2 second coefficient multiple，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Euclid step2 first coefficient difference，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Euclid step2 first coefficient multiple，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Euclid step2 remainder，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Euclid step2 quotient times divisor，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Euclid step2 exact ratio，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Euclid step1 second coefficient difference，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Euclid step1 second coefficient multiple，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Euclid step1 first coefficient difference，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Euclid step1 first coefficient multiple，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Euclid step1 remainder，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Euclid step1 quotient times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Euclid step1 exact ratio，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 以精確有理數重算 Euclid step0 second coefficient difference，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Euclid step0 second coefficient multiple，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Euclid step0 first coefficient difference，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Euclid step0 first coefficient multiple，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Euclid step0 remainder，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Euclid step0 quotient times divisor，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Euclid step0 exact ratio，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(1, -1, 6, 1, 是, extended_euclidean_back_substitution)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。