第 18802 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":-424,"b":91}。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":-424,"b":91}。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=-424,b=91

initial nonnegative chain values=[424, 91];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(424, 91)

Euclid step0 exact ratio = 424/91

Euclid step0 quotient times divisor := mul(4, 91)

Euclid step0 quotient times divisor = 364

Euclid step0 remainder := sub(424, 364)

Euclid step0 remainder = 60

step=0;left=424;positive right=91;exact ratio=424/91;floor quotient=4

left=right*q+r with r=60 in[0,91);gcd pair=[1, 1]

Euclid step0 first coefficient multiple := mul(4, 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(4, 1)

Euclid step0 second coefficient multiple = 4

Euclid step0 second coefficient difference := sub(0, 4)

Euclid step0 second coefficient difference = -4

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=4

next remainder coefficient pair=[1, -4];abs(a)*s_next+abs(b)*t_next=60

Euclid step1 exact ratio := div(91, 60)

Euclid step1 exact ratio = 91/60

Euclid step1 quotient times divisor := mul(1, 60)

Euclid step1 quotient times divisor = 60

Euclid step1 remainder := sub(91, 60)

Euclid step1 remainder = 31

step=1;left=91;positive right=60;exact ratio=91/60;floor quotient=1

left=right*q+r with r=31 in[0,60);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, -4)

Euclid step1 second coefficient multiple = -4

Euclid step1 second coefficient difference := sub(1, -4)

Euclid step1 second coefficient difference = 5

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -4]];same quotient=1

next remainder coefficient pair=[-1, 5];abs(a)*s_next+abs(b)*t_next=31

Euclid step2 exact ratio := div(60, 31)

Euclid step2 exact ratio = 60/31

Euclid step2 quotient times divisor := mul(1, 31)

Euclid step2 quotient times divisor = 31

Euclid step2 remainder := sub(60, 31)

Euclid step2 remainder = 29

step=2;left=60;positive right=31;exact ratio=60/31;floor quotient=1

left=right*q+r with r=29 in[0,31);gcd pair=[1, 1]

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

Euclid step2 first coefficient multiple = -1

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

Euclid step2 first coefficient difference = 2

Euclid step2 second coefficient multiple := mul(1, 5)

Euclid step2 second coefficient multiple = 5

Euclid step2 second coefficient difference := sub(-4, 5)

Euclid step2 second coefficient difference = -9

step=2;two original-absolute-input coefficient pairs=[[1, -4], [-1, 5]];same quotient=1

next remainder coefficient pair=[2, -9];abs(a)*s_next+abs(b)*t_next=29

Euclid step3 exact ratio := div(31, 29)

Euclid step3 exact ratio = 31/29

Euclid step3 quotient times divisor := mul(1, 29)

Euclid step3 quotient times divisor = 29

Euclid step3 remainder := sub(31, 29)

Euclid step3 remainder = 2

step=3;left=31;positive right=29;exact ratio=31/29;floor quotient=1

left=right*q+r with r=2 in[0,29);gcd pair=[1, 1]

Euclid step3 first coefficient multiple := mul(1, 2)

Euclid step3 first coefficient multiple = 2

Euclid step3 first coefficient difference := sub(-1, 2)

Euclid step3 first coefficient difference = -3

Euclid step3 second coefficient multiple := mul(1, -9)

Euclid step3 second coefficient multiple = -9

Euclid step3 second coefficient difference := sub(5, -9)

Euclid step3 second coefficient difference = 14

step=3;two original-absolute-input coefficient pairs=[[-1, 5], [2, -9]];same quotient=1

next remainder coefficient pair=[-3, 14];abs(a)*s_next+abs(b)*t_next=2

Euclid step4 exact ratio := div(29, 2)

Euclid step4 exact ratio = 29/2

Euclid step4 quotient times divisor := mul(14, 2)

Euclid step4 quotient times divisor = 28

Euclid step4 remainder := sub(29, 28)

Euclid step4 remainder = 1

step=4;left=29;positive right=2;exact ratio=29/2;floor quotient=14

left=right*q+r with r=1 in[0,2);gcd pair=[1, 1]

Euclid step4 first coefficient multiple := mul(14, -3)

Euclid step4 first coefficient multiple = -42

Euclid step4 first coefficient difference := sub(2, -42)

Euclid step4 first coefficient difference = 44

Euclid step4 second coefficient multiple := mul(14, 14)

Euclid step4 second coefficient multiple = 196

Euclid step4 second coefficient difference := sub(-9, 196)

Euclid step4 second coefficient difference = -205

step=4;two original-absolute-input coefficient pairs=[[2, -9], [-3, 14]];same quotient=14

next remainder coefficient pair=[44, -205];abs(a)*s_next+abs(b)*t_next=1

Euclid step5 exact ratio := div(2, 1)

Euclid step5 exact ratio = 2

Euclid step5 quotient times divisor := mul(2, 1)

Euclid step5 quotient times divisor = 2

Euclid step5 remainder := sub(2, 2)

Euclid step5 remainder = 0

step=5;left=2;positive right=1;exact ratio=2;floor quotient=2

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

Euclid step5 first coefficient multiple := mul(2, 44)

Euclid step5 first coefficient multiple = 88

Euclid step5 first coefficient difference := sub(-3, 88)

Euclid step5 first coefficient difference = -91

Euclid step5 second coefficient multiple := mul(2, -205)

Euclid step5 second coefficient multiple = -410

Euclid step5 second coefficient difference := sub(14, -410)

Euclid step5 second coefficient difference = 424

step=5;two original-absolute-input coefficient pairs=[[-3, 14], [44, -205]];same quotient=2

next remainder coefficient pair=[-91, 424];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[424, 91, 4, 60], [91, 60, 1, 31], [60, 31, 1, 29], [31, 29, 1, 2], [29, 2, 14, 1], [2, 1, 2, 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, 44)

Euclid restore original a sign in coefficient = -44

Euclid restore original b sign in coefficient := mul(1, -205)

Euclid restore original b sign in coefficient = -205

Euclid ORIGINAL signed a times x := mul(-424, -44)

Euclid ORIGINAL signed a times x = 18656

Euclid ORIGINAL signed b times y := mul(91, -205)

Euclid ORIGINAL signed b times y = -18655

Euclid final signed Bezout linear combination := sum(18656, -18655)

Euclid final signed Bezout linear combination = 1

original a=-424,b=91;deterministic absolute-chain coefficients=[44, -205]

sign-restored source coefficients x=-44,y=-205;a*x+b*y=1=nonnegative gcd

逐步反向核對

• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · premise_and_result_audit — 以精確有理數重算 Euclid step5 second coefficient difference，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Euclid step5 second coefficient multiple，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euclid step5 first coefficient difference，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euclid step5 first coefficient multiple，不以小數近似替代。
• S051 · 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。
• S050 · premise_and_result_audit — 以精確有理數重算 Euclid step5 remainder，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euclid step5 quotient times divisor，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Euclid step5 exact ratio，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Euclid step4 second coefficient difference，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euclid step4 second coefficient multiple，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euclid step4 first coefficient difference，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Euclid step4 first coefficient multiple，不以小數近似替代。
• S042 · 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。
• S041 · premise_and_result_audit — 以精確有理數重算 Euclid step4 remainder，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euclid step4 quotient times divisor，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euclid step4 exact ratio，不以小數近似替代。
• 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

(1, -44, -205, 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。