第 18818 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":-792,"b":123}。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":-792,"b":123}。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=-792,b=123

initial nonnegative chain values=[792, 123];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(792, 123)

Euclid step0 exact ratio = 264/41

Euclid step0 quotient times divisor := mul(6, 123)

Euclid step0 quotient times divisor = 738

Euclid step0 remainder := sub(792, 738)

Euclid step0 remainder = 54

step=0;left=792;positive right=123;exact ratio=264/41;floor quotient=6

left=right*q+r with r=54 in[0,123);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=54

Euclid step1 exact ratio := div(123, 54)

Euclid step1 exact ratio = 41/18

Euclid step1 quotient times divisor := mul(2, 54)

Euclid step1 quotient times divisor = 108

Euclid step1 remainder := sub(123, 108)

Euclid step1 remainder = 15

step=1;left=123;positive right=54;exact ratio=41/18;floor quotient=2

left=right*q+r with r=15 in[0,54);gcd pair=[3, 3]

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=15

Euclid step2 exact ratio := div(54, 15)

Euclid step2 exact ratio = 18/5

Euclid step2 quotient times divisor := mul(3, 15)

Euclid step2 quotient times divisor = 45

Euclid step2 remainder := sub(54, 45)

Euclid step2 remainder = 9

step=2;left=54;positive right=15;exact ratio=18/5;floor quotient=3

left=right*q+r with r=9 in[0,15);gcd pair=[3, 3]

Euclid step2 first coefficient multiple := mul(3, -2)

Euclid step2 first coefficient multiple = -6

Euclid step2 first coefficient difference := sub(1, -6)

Euclid step2 first coefficient difference = 7

Euclid step2 second coefficient multiple := mul(3, 13)

Euclid step2 second coefficient multiple = 39

Euclid step2 second coefficient difference := sub(-6, 39)

Euclid step2 second coefficient difference = -45

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-2, 13]];same quotient=3

next remainder coefficient pair=[7, -45];abs(a)*s_next+abs(b)*t_next=9

Euclid step3 exact ratio := div(15, 9)

Euclid step3 exact ratio = 5/3

Euclid step3 quotient times divisor := mul(1, 9)

Euclid step3 quotient times divisor = 9

Euclid step3 remainder := sub(15, 9)

Euclid step3 remainder = 6

step=3;left=15;positive right=9;exact ratio=5/3;floor quotient=1

left=right*q+r with r=6 in[0,9);gcd pair=[3, 3]

Euclid step3 first coefficient multiple := mul(1, 7)

Euclid step3 first coefficient multiple = 7

Euclid step3 first coefficient difference := sub(-2, 7)

Euclid step3 first coefficient difference = -9

Euclid step3 second coefficient multiple := mul(1, -45)

Euclid step3 second coefficient multiple = -45

Euclid step3 second coefficient difference := sub(13, -45)

Euclid step3 second coefficient difference = 58

step=3;two original-absolute-input coefficient pairs=[[-2, 13], [7, -45]];same quotient=1

next remainder coefficient pair=[-9, 58];abs(a)*s_next+abs(b)*t_next=6

Euclid step4 exact ratio := div(9, 6)

Euclid step4 exact ratio = 3/2

Euclid step4 quotient times divisor := mul(1, 6)

Euclid step4 quotient times divisor = 6

Euclid step4 remainder := sub(9, 6)

Euclid step4 remainder = 3

step=4;left=9;positive right=6;exact ratio=3/2;floor quotient=1

left=right*q+r with r=3 in[0,6);gcd pair=[3, 3]

Euclid step4 first coefficient multiple := mul(1, -9)

Euclid step4 first coefficient multiple = -9

Euclid step4 first coefficient difference := sub(7, -9)

Euclid step4 first coefficient difference = 16

Euclid step4 second coefficient multiple := mul(1, 58)

Euclid step4 second coefficient multiple = 58

Euclid step4 second coefficient difference := sub(-45, 58)

Euclid step4 second coefficient difference = -103

step=4;two original-absolute-input coefficient pairs=[[7, -45], [-9, 58]];same quotient=1

next remainder coefficient pair=[16, -103];abs(a)*s_next+abs(b)*t_next=3

Euclid step5 exact ratio := div(6, 3)

Euclid step5 exact ratio = 2

Euclid step5 quotient times divisor := mul(2, 3)

Euclid step5 quotient times divisor = 6

Euclid step5 remainder := sub(6, 6)

Euclid step5 remainder = 0

step=5;left=6;positive right=3;exact ratio=2;floor quotient=2

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

Euclid step5 first coefficient multiple := mul(2, 16)

Euclid step5 first coefficient multiple = 32

Euclid step5 first coefficient difference := sub(-9, 32)

Euclid step5 first coefficient difference = -41

Euclid step5 second coefficient multiple := mul(2, -103)

Euclid step5 second coefficient multiple = -206

Euclid step5 second coefficient difference := sub(58, -206)

Euclid step5 second coefficient difference = 264

step=5;two original-absolute-input coefficient pairs=[[-9, 58], [16, -103]];same quotient=2

next remainder coefficient pair=[-41, 264];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[792, 123, 6, 54], [123, 54, 2, 15], [54, 15, 3, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 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, 16)

Euclid restore original a sign in coefficient = -16

Euclid restore original b sign in coefficient := mul(1, -103)

Euclid restore original b sign in coefficient = -103

Euclid ORIGINAL signed a times x := mul(-792, -16)

Euclid ORIGINAL signed a times x = 12672

Euclid ORIGINAL signed b times y := mul(123, -103)

Euclid ORIGINAL signed b times y = -12669

Euclid final signed Bezout linear combination := sum(12672, -12669)

Euclid final signed Bezout linear combination = 3

original a=-792,b=123;deterministic absolute-chain coefficients=[16, -103]

sign-restored source coefficients x=-16,y=-103;a*x+b*y=3=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

(3, -16, -103, 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。