第 18817 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":769,"b":121}。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":769,"b":121}。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=769,b=121

initial nonnegative chain values=[769, 121];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(769, 121)

Euclid step0 exact ratio = 769/121

Euclid step0 quotient times divisor := mul(6, 121)

Euclid step0 quotient times divisor = 726

Euclid step0 remainder := sub(769, 726)

Euclid step0 remainder = 43

step=0;left=769;positive right=121;exact ratio=769/121;floor quotient=6

left=right*q+r with r=43 in[0,121);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=43

Euclid step1 exact ratio := div(121, 43)

Euclid step1 exact ratio = 121/43

Euclid step1 quotient times divisor := mul(2, 43)

Euclid step1 quotient times divisor = 86

Euclid step1 remainder := sub(121, 86)

Euclid step1 remainder = 35

step=1;left=121;positive right=43;exact ratio=121/43;floor quotient=2

left=right*q+r with r=35 in[0,43);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, -6)

Euclid step1 second coefficient multiple = -12

Euclid step1 second coefficient difference := sub(1, -12)

Euclid step1 second coefficient difference = 13

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=2

next remainder coefficient pair=[-2, 13];abs(a)*s_next+abs(b)*t_next=35

Euclid step2 exact ratio := div(43, 35)

Euclid step2 exact ratio = 43/35

Euclid step2 quotient times divisor := mul(1, 35)

Euclid step2 quotient times divisor = 35

Euclid step2 remainder := sub(43, 35)

Euclid step2 remainder = 8

step=2;left=43;positive right=35;exact ratio=43/35;floor quotient=1

left=right*q+r with r=8 in[0,35);gcd pair=[1, 1]

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

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(1, 13)

Euclid step2 second coefficient multiple = 13

Euclid step2 second coefficient difference := sub(-6, 13)

Euclid step2 second coefficient difference = -19

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-2, 13]];same quotient=1

next remainder coefficient pair=[3, -19];abs(a)*s_next+abs(b)*t_next=8

Euclid step3 exact ratio := div(35, 8)

Euclid step3 exact ratio = 35/8

Euclid step3 quotient times divisor := mul(4, 8)

Euclid step3 quotient times divisor = 32

Euclid step3 remainder := sub(35, 32)

Euclid step3 remainder = 3

step=3;left=35;positive right=8;exact ratio=35/8;floor quotient=4

left=right*q+r with r=3 in[0,8);gcd pair=[1, 1]

Euclid step3 first coefficient multiple := mul(4, 3)

Euclid step3 first coefficient multiple = 12

Euclid step3 first coefficient difference := sub(-2, 12)

Euclid step3 first coefficient difference = -14

Euclid step3 second coefficient multiple := mul(4, -19)

Euclid step3 second coefficient multiple = -76

Euclid step3 second coefficient difference := sub(13, -76)

Euclid step3 second coefficient difference = 89

step=3;two original-absolute-input coefficient pairs=[[-2, 13], [3, -19]];same quotient=4

next remainder coefficient pair=[-14, 89];abs(a)*s_next+abs(b)*t_next=3

Euclid step4 exact ratio := div(8, 3)

Euclid step4 exact ratio = 8/3

Euclid step4 quotient times divisor := mul(2, 3)

Euclid step4 quotient times divisor = 6

Euclid step4 remainder := sub(8, 6)

Euclid step4 remainder = 2

step=4;left=8;positive right=3;exact ratio=8/3;floor quotient=2

left=right*q+r with r=2 in[0,3);gcd pair=[1, 1]

Euclid step4 first coefficient multiple := mul(2, -14)

Euclid step4 first coefficient multiple = -28

Euclid step4 first coefficient difference := sub(3, -28)

Euclid step4 first coefficient difference = 31

Euclid step4 second coefficient multiple := mul(2, 89)

Euclid step4 second coefficient multiple = 178

Euclid step4 second coefficient difference := sub(-19, 178)

Euclid step4 second coefficient difference = -197

step=4;two original-absolute-input coefficient pairs=[[3, -19], [-14, 89]];same quotient=2

next remainder coefficient pair=[31, -197];abs(a)*s_next+abs(b)*t_next=2

Euclid step5 exact ratio := div(3, 2)

Euclid step5 exact ratio = 3/2

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

Euclid step5 quotient times divisor = 2

Euclid step5 remainder := sub(3, 2)

Euclid step5 remainder = 1

step=5;left=3;positive right=2;exact ratio=3/2;floor quotient=1

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

Euclid step5 first coefficient multiple := mul(1, 31)

Euclid step5 first coefficient multiple = 31

Euclid step5 first coefficient difference := sub(-14, 31)

Euclid step5 first coefficient difference = -45

Euclid step5 second coefficient multiple := mul(1, -197)

Euclid step5 second coefficient multiple = -197

Euclid step5 second coefficient difference := sub(89, -197)

Euclid step5 second coefficient difference = 286

step=5;two original-absolute-input coefficient pairs=[[-14, 89], [31, -197]];same quotient=1

next remainder coefficient pair=[-45, 286];abs(a)*s_next+abs(b)*t_next=1

Euclid step6 exact ratio := div(2, 1)

Euclid step6 exact ratio = 2

Euclid step6 quotient times divisor := mul(2, 1)

Euclid step6 quotient times divisor = 2

Euclid step6 remainder := sub(2, 2)

Euclid step6 remainder = 0

step=6;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 step6 first coefficient multiple := mul(2, -45)

Euclid step6 first coefficient multiple = -90

Euclid step6 first coefficient difference := sub(31, -90)

Euclid step6 first coefficient difference = 121

Euclid step6 second coefficient multiple := mul(2, 286)

Euclid step6 second coefficient multiple = 572

Euclid step6 second coefficient difference := sub(-197, 572)

Euclid step6 second coefficient difference = -769

step=6;two original-absolute-input coefficient pairs=[[31, -197], [-45, 286]];same quotient=2

next remainder coefficient pair=[121, -769];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[769, 121, 6, 43], [121, 43, 2, 35], [43, 35, 1, 8], [35, 8, 4, 3], [8, 3, 2, 2], [3, 2, 1, 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, -45)

Euclid restore original a sign in coefficient = -45

Euclid restore original b sign in coefficient := mul(1, 286)

Euclid restore original b sign in coefficient = 286

Euclid ORIGINAL signed a times x := mul(769, -45)

Euclid ORIGINAL signed a times x = -34605

Euclid ORIGINAL signed b times y := mul(121, 286)

Euclid ORIGINAL signed b times y = 34606

Euclid final signed Bezout linear combination := sum(-34605, 34606)

Euclid final signed Bezout linear combination = 1

original a=769,b=121;deterministic absolute-chain coefficients=[-45, 286]

sign-restored source coefficients x=-45,y=286;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, -45, 286, 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。