第 18821 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":861,"b":129}。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":861,"b":129}。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=861,b=129

initial nonnegative chain values=[861, 129];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(861, 129)

Euclid step0 exact ratio = 287/43

Euclid step0 quotient times divisor := mul(6, 129)

Euclid step0 quotient times divisor = 774

Euclid step0 remainder := sub(861, 774)

Euclid step0 remainder = 87

step=0;left=861;positive right=129;exact ratio=287/43;floor quotient=6

left=right*q+r with r=87 in[0,129);gcd pair=[3, 3]

Euclid step0 first coefficient multiple := mul(6, 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(6, 1)

Euclid step0 second coefficient multiple = 6

Euclid step0 second coefficient difference := sub(0, 6)

Euclid step0 second coefficient difference = -6

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=6

next remainder coefficient pair=[1, -6];abs(a)*s_next+abs(b)*t_next=87

Euclid step1 exact ratio := div(129, 87)

Euclid step1 exact ratio = 43/29

Euclid step1 quotient times divisor := mul(1, 87)

Euclid step1 quotient times divisor = 87

Euclid step1 remainder := sub(129, 87)

Euclid step1 remainder = 42

step=1;left=129;positive right=87;exact ratio=43/29;floor quotient=1

left=right*q+r with r=42 in[0,87);gcd pair=[3, 3]

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, -6)

Euclid step1 second coefficient multiple = -6

Euclid step1 second coefficient difference := sub(1, -6)

Euclid step1 second coefficient difference = 7

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=1

next remainder coefficient pair=[-1, 7];abs(a)*s_next+abs(b)*t_next=42

Euclid step2 exact ratio := div(87, 42)

Euclid step2 exact ratio = 29/14

Euclid step2 quotient times divisor := mul(2, 42)

Euclid step2 quotient times divisor = 84

Euclid step2 remainder := sub(87, 84)

Euclid step2 remainder = 3

step=2;left=87;positive right=42;exact ratio=29/14;floor quotient=2

left=right*q+r with r=3 in[0,42);gcd pair=[3, 3]

Euclid step2 first coefficient multiple := mul(2, -1)

Euclid step2 first coefficient multiple = -2

Euclid step2 first coefficient difference := sub(1, -2)

Euclid step2 first coefficient difference = 3

Euclid step2 second coefficient multiple := mul(2, 7)

Euclid step2 second coefficient multiple = 14

Euclid step2 second coefficient difference := sub(-6, 14)

Euclid step2 second coefficient difference = -20

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-1, 7]];same quotient=2

next remainder coefficient pair=[3, -20];abs(a)*s_next+abs(b)*t_next=3

Euclid step3 exact ratio := div(42, 3)

Euclid step3 exact ratio = 14

Euclid step3 quotient times divisor := mul(14, 3)

Euclid step3 quotient times divisor = 42

Euclid step3 remainder := sub(42, 42)

Euclid step3 remainder = 0

step=3;left=42;positive right=3;exact ratio=14;floor quotient=14

left=right*q+r with r=0 in[0,3);gcd pair=[3, 3]

Euclid step3 first coefficient multiple := mul(14, 3)

Euclid step3 first coefficient multiple = 42

Euclid step3 first coefficient difference := sub(-1, 42)

Euclid step3 first coefficient difference = -43

Euclid step3 second coefficient multiple := mul(14, -20)

Euclid step3 second coefficient multiple = -280

Euclid step3 second coefficient difference := sub(7, -280)

Euclid step3 second coefficient difference = 287

step=3;two original-absolute-input coefficient pairs=[[-1, 7], [3, -20]];same quotient=14

next remainder coefficient pair=[-43, 287];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[861, 129, 6, 87], [129, 87, 1, 42], [87, 42, 2, 3], [42, 3, 14, 0]];terminal pair=[3, 0]

gcd(a,b)=gcd(3,0)=3;all divisions used positive divisors;nonnegative gcd convention

Euclid restore original a sign in coefficient := mul(1, 3)

Euclid restore original a sign in coefficient = 3

Euclid restore original b sign in coefficient := mul(1, -20)

Euclid restore original b sign in coefficient = -20

Euclid ORIGINAL signed a times x := mul(861, 3)

Euclid ORIGINAL signed a times x = 2583

Euclid ORIGINAL signed b times y := mul(129, -20)

Euclid ORIGINAL signed b times y = -2580

Euclid final signed Bezout linear combination := sum(2583, -2580)

Euclid final signed Bezout linear combination = 3

original a=861,b=129;deterministic absolute-chain coefficients=[3, -20]

sign-restored source coefficients x=3,y=-20;a*x+b*y=3=nonnegative gcd

逐步反向核對

• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Euclid step3 second coefficient difference，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euclid step3 second coefficient multiple，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euclid step3 first coefficient difference，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euclid step3 first coefficient multiple，不以小數近似替代。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Euclid step3 remainder，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Euclid step3 quotient times divisor，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Euclid step3 exact ratio，不以小數近似替代。
• 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

(3, 3, -20, 3, 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。