第 18816 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":-746,"b":119}。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":-746,"b":119}。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=-746,b=119

initial nonnegative chain values=[746, 119];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(746, 119)

Euclid step0 exact ratio = 746/119

Euclid step0 quotient times divisor := mul(6, 119)

Euclid step0 quotient times divisor = 714

Euclid step0 remainder := sub(746, 714)

Euclid step0 remainder = 32

step=0;left=746;positive right=119;exact ratio=746/119;floor quotient=6

left=right*q+r with r=32 in[0,119);gcd pair=[1, 1]

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=32

Euclid step1 exact ratio := div(119, 32)

Euclid step1 exact ratio = 119/32

Euclid step1 quotient times divisor := mul(3, 32)

Euclid step1 quotient times divisor = 96

Euclid step1 remainder := sub(119, 96)

Euclid step1 remainder = 23

step=1;left=119;positive right=32;exact ratio=119/32;floor quotient=3

left=right*q+r with r=23 in[0,32);gcd pair=[1, 1]

Euclid step1 first coefficient multiple := mul(3, 1)

Euclid step1 first coefficient multiple = 3

Euclid step1 first coefficient difference := sub(0, 3)

Euclid step1 first coefficient difference = -3

Euclid step1 second coefficient multiple := mul(3, -6)

Euclid step1 second coefficient multiple = -18

Euclid step1 second coefficient difference := sub(1, -18)

Euclid step1 second coefficient difference = 19

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=3

next remainder coefficient pair=[-3, 19];abs(a)*s_next+abs(b)*t_next=23

Euclid step2 exact ratio := div(32, 23)

Euclid step2 exact ratio = 32/23

Euclid step2 quotient times divisor := mul(1, 23)

Euclid step2 quotient times divisor = 23

Euclid step2 remainder := sub(32, 23)

Euclid step2 remainder = 9

step=2;left=32;positive right=23;exact ratio=32/23;floor quotient=1

left=right*q+r with r=9 in[0,23);gcd pair=[1, 1]

Euclid step2 first coefficient multiple := mul(1, -3)

Euclid step2 first coefficient multiple = -3

Euclid step2 first coefficient difference := sub(1, -3)

Euclid step2 first coefficient difference = 4

Euclid step2 second coefficient multiple := mul(1, 19)

Euclid step2 second coefficient multiple = 19

Euclid step2 second coefficient difference := sub(-6, 19)

Euclid step2 second coefficient difference = -25

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-3, 19]];same quotient=1

next remainder coefficient pair=[4, -25];abs(a)*s_next+abs(b)*t_next=9

Euclid step3 exact ratio := div(23, 9)

Euclid step3 exact ratio = 23/9

Euclid step3 quotient times divisor := mul(2, 9)

Euclid step3 quotient times divisor = 18

Euclid step3 remainder := sub(23, 18)

Euclid step3 remainder = 5

step=3;left=23;positive right=9;exact ratio=23/9;floor quotient=2

left=right*q+r with r=5 in[0,9);gcd pair=[1, 1]

Euclid step3 first coefficient multiple := mul(2, 4)

Euclid step3 first coefficient multiple = 8

Euclid step3 first coefficient difference := sub(-3, 8)

Euclid step3 first coefficient difference = -11

Euclid step3 second coefficient multiple := mul(2, -25)

Euclid step3 second coefficient multiple = -50

Euclid step3 second coefficient difference := sub(19, -50)

Euclid step3 second coefficient difference = 69

step=3;two original-absolute-input coefficient pairs=[[-3, 19], [4, -25]];same quotient=2

next remainder coefficient pair=[-11, 69];abs(a)*s_next+abs(b)*t_next=5

Euclid step4 exact ratio := div(9, 5)

Euclid step4 exact ratio = 9/5

Euclid step4 quotient times divisor := mul(1, 5)

Euclid step4 quotient times divisor = 5

Euclid step4 remainder := sub(9, 5)

Euclid step4 remainder = 4

step=4;left=9;positive right=5;exact ratio=9/5;floor quotient=1

left=right*q+r with r=4 in[0,5);gcd pair=[1, 1]

Euclid step4 first coefficient multiple := mul(1, -11)

Euclid step4 first coefficient multiple = -11

Euclid step4 first coefficient difference := sub(4, -11)

Euclid step4 first coefficient difference = 15

Euclid step4 second coefficient multiple := mul(1, 69)

Euclid step4 second coefficient multiple = 69

Euclid step4 second coefficient difference := sub(-25, 69)

Euclid step4 second coefficient difference = -94

step=4;two original-absolute-input coefficient pairs=[[4, -25], [-11, 69]];same quotient=1

next remainder coefficient pair=[15, -94];abs(a)*s_next+abs(b)*t_next=4

Euclid step5 exact ratio := div(5, 4)

Euclid step5 exact ratio = 5/4

Euclid step5 quotient times divisor := mul(1, 4)

Euclid step5 quotient times divisor = 4

Euclid step5 remainder := sub(5, 4)

Euclid step5 remainder = 1

step=5;left=5;positive right=4;exact ratio=5/4;floor quotient=1

left=right*q+r with r=1 in[0,4);gcd pair=[1, 1]

Euclid step5 first coefficient multiple := mul(1, 15)

Euclid step5 first coefficient multiple = 15

Euclid step5 first coefficient difference := sub(-11, 15)

Euclid step5 first coefficient difference = -26

Euclid step5 second coefficient multiple := mul(1, -94)

Euclid step5 second coefficient multiple = -94

Euclid step5 second coefficient difference := sub(69, -94)

Euclid step5 second coefficient difference = 163

step=5;two original-absolute-input coefficient pairs=[[-11, 69], [15, -94]];same quotient=1

next remainder coefficient pair=[-26, 163];abs(a)*s_next+abs(b)*t_next=1

Euclid step6 exact ratio := div(4, 1)

Euclid step6 exact ratio = 4

Euclid step6 quotient times divisor := mul(4, 1)

Euclid step6 quotient times divisor = 4

Euclid step6 remainder := sub(4, 4)

Euclid step6 remainder = 0

step=6;left=4;positive right=1;exact ratio=4;floor quotient=4

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

Euclid step6 first coefficient multiple := mul(4, -26)

Euclid step6 first coefficient multiple = -104

Euclid step6 first coefficient difference := sub(15, -104)

Euclid step6 first coefficient difference = 119

Euclid step6 second coefficient multiple := mul(4, 163)

Euclid step6 second coefficient multiple = 652

Euclid step6 second coefficient difference := sub(-94, 652)

Euclid step6 second coefficient difference = -746

step=6;two original-absolute-input coefficient pairs=[[15, -94], [-26, 163]];same quotient=4

next remainder coefficient pair=[119, -746];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[746, 119, 6, 32], [119, 32, 3, 23], [32, 23, 1, 9], [23, 9, 2, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 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, -26)

Euclid restore original a sign in coefficient = 26

Euclid restore original b sign in coefficient := mul(1, 163)

Euclid restore original b sign in coefficient = 163

Euclid ORIGINAL signed a times x := mul(-746, 26)

Euclid ORIGINAL signed a times x = -19396

Euclid ORIGINAL signed b times y := mul(119, 163)

Euclid ORIGINAL signed b times y = 19397

Euclid final signed Bezout linear combination := sum(-19396, 19397)

Euclid final signed Bezout linear combination = 1

original a=-746,b=119;deterministic absolute-chain coefficients=[-26, 163]

sign-restored source coefficients x=26,y=163;a*x+b*y=1=nonnegative gcd

逐步反向核對

• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · premise_and_result_audit — 以精確有理數重算 Euclid step6 second coefficient difference，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Euclid step6 second coefficient multiple，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Euclid step6 first coefficient difference，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Euclid step6 first coefficient multiple，不以小數近似替代。
• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 Euclid step6 remainder，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Euclid step6 quotient times divisor，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Euclid step6 exact ratio，不以小數近似替代。
• 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, 26, 163, 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。