第 18808 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":-562,"b":103}。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":-562,"b":103}。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=-562,b=103

initial nonnegative chain values=[562, 103];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(562, 103)

Euclid step0 exact ratio = 562/103

Euclid step0 quotient times divisor := mul(5, 103)

Euclid step0 quotient times divisor = 515

Euclid step0 remainder := sub(562, 515)

Euclid step0 remainder = 47

step=0;left=562;positive right=103;exact ratio=562/103;floor quotient=5

left=right*q+r with r=47 in[0,103);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=47

Euclid step1 exact ratio := div(103, 47)

Euclid step1 exact ratio = 103/47

Euclid step1 quotient times divisor := mul(2, 47)

Euclid step1 quotient times divisor = 94

Euclid step1 remainder := sub(103, 94)

Euclid step1 remainder = 9

step=1;left=103;positive right=47;exact ratio=103/47;floor quotient=2

left=right*q+r with r=9 in[0,47);gcd pair=[1, 1]

Euclid step1 first coefficient multiple := mul(2, 1)

Euclid step1 first coefficient multiple = 2

Euclid step1 first coefficient difference := sub(0, 2)

Euclid step1 first coefficient difference = -2

Euclid step1 second coefficient multiple := mul(2, -5)

Euclid step1 second coefficient multiple = -10

Euclid step1 second coefficient difference := sub(1, -10)

Euclid step1 second coefficient difference = 11

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -5]];same quotient=2

next remainder coefficient pair=[-2, 11];abs(a)*s_next+abs(b)*t_next=9

Euclid step2 exact ratio := div(47, 9)

Euclid step2 exact ratio = 47/9

Euclid step2 quotient times divisor := mul(5, 9)

Euclid step2 quotient times divisor = 45

Euclid step2 remainder := sub(47, 45)

Euclid step2 remainder = 2

step=2;left=47;positive right=9;exact ratio=47/9;floor quotient=5

left=right*q+r with r=2 in[0,9);gcd pair=[1, 1]

Euclid step2 first coefficient multiple := mul(5, -2)

Euclid step2 first coefficient multiple = -10

Euclid step2 first coefficient difference := sub(1, -10)

Euclid step2 first coefficient difference = 11

Euclid step2 second coefficient multiple := mul(5, 11)

Euclid step2 second coefficient multiple = 55

Euclid step2 second coefficient difference := sub(-5, 55)

Euclid step2 second coefficient difference = -60

step=2;two original-absolute-input coefficient pairs=[[1, -5], [-2, 11]];same quotient=5

next remainder coefficient pair=[11, -60];abs(a)*s_next+abs(b)*t_next=2

Euclid step3 exact ratio := div(9, 2)

Euclid step3 exact ratio = 9/2

Euclid step3 quotient times divisor := mul(4, 2)

Euclid step3 quotient times divisor = 8

Euclid step3 remainder := sub(9, 8)

Euclid step3 remainder = 1

step=3;left=9;positive right=2;exact ratio=9/2;floor quotient=4

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

Euclid step3 first coefficient multiple := mul(4, 11)

Euclid step3 first coefficient multiple = 44

Euclid step3 first coefficient difference := sub(-2, 44)

Euclid step3 first coefficient difference = -46

Euclid step3 second coefficient multiple := mul(4, -60)

Euclid step3 second coefficient multiple = -240

Euclid step3 second coefficient difference := sub(11, -240)

Euclid step3 second coefficient difference = 251

step=3;two original-absolute-input coefficient pairs=[[-2, 11], [11, -60]];same quotient=4

next remainder coefficient pair=[-46, 251];abs(a)*s_next+abs(b)*t_next=1

Euclid step4 exact ratio := div(2, 1)

Euclid step4 exact ratio = 2

Euclid step4 quotient times divisor := mul(2, 1)

Euclid step4 quotient times divisor = 2

Euclid step4 remainder := sub(2, 2)

Euclid step4 remainder = 0

step=4;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 step4 first coefficient multiple := mul(2, -46)

Euclid step4 first coefficient multiple = -92

Euclid step4 first coefficient difference := sub(11, -92)

Euclid step4 first coefficient difference = 103

Euclid step4 second coefficient multiple := mul(2, 251)

Euclid step4 second coefficient multiple = 502

Euclid step4 second coefficient difference := sub(-60, 502)

Euclid step4 second coefficient difference = -562

step=4;two original-absolute-input coefficient pairs=[[11, -60], [-46, 251]];same quotient=2

next remainder coefficient pair=[103, -562];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[562, 103, 5, 47], [103, 47, 2, 9], [47, 9, 5, 2], [9, 2, 4, 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, -46)

Euclid restore original a sign in coefficient = 46

Euclid restore original b sign in coefficient := mul(1, 251)

Euclid restore original b sign in coefficient = 251

Euclid ORIGINAL signed a times x := mul(-562, 46)

Euclid ORIGINAL signed a times x = -25852

Euclid ORIGINAL signed b times y := mul(103, 251)

Euclid ORIGINAL signed b times y = 25853

Euclid final signed Bezout linear combination := sum(-25852, 25853)

Euclid final signed Bezout linear combination = 1

original a=-562,b=103;deterministic absolute-chain coefficients=[-46, 251]

sign-restored source coefficients x=46,y=251;a*x+b*y=1=nonnegative gcd

逐步反向核對

• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• 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, 46, 251, 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。