第 18812 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":-654,"b":111}。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":-654,"b":111}。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=-654,b=111

initial nonnegative chain values=[654, 111];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(654, 111)

Euclid step0 exact ratio = 218/37

Euclid step0 quotient times divisor := mul(5, 111)

Euclid step0 quotient times divisor = 555

Euclid step0 remainder := sub(654, 555)

Euclid step0 remainder = 99

step=0;left=654;positive right=111;exact ratio=218/37;floor quotient=5

left=right*q+r with r=99 in[0,111);gcd pair=[3, 3]

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=99

Euclid step1 exact ratio := div(111, 99)

Euclid step1 exact ratio = 37/33

Euclid step1 quotient times divisor := mul(1, 99)

Euclid step1 quotient times divisor = 99

Euclid step1 remainder := sub(111, 99)

Euclid step1 remainder = 12

step=1;left=111;positive right=99;exact ratio=37/33;floor quotient=1

left=right*q+r with r=12 in[0,99);gcd pair=[3, 3]

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, -5)

Euclid step1 second coefficient multiple = -5

Euclid step1 second coefficient difference := sub(1, -5)

Euclid step1 second coefficient difference = 6

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

next remainder coefficient pair=[-1, 6];abs(a)*s_next+abs(b)*t_next=12

Euclid step2 exact ratio := div(99, 12)

Euclid step2 exact ratio = 33/4

Euclid step2 quotient times divisor := mul(8, 12)

Euclid step2 quotient times divisor = 96

Euclid step2 remainder := sub(99, 96)

Euclid step2 remainder = 3

step=2;left=99;positive right=12;exact ratio=33/4;floor quotient=8

left=right*q+r with r=3 in[0,12);gcd pair=[3, 3]

Euclid step2 first coefficient multiple := mul(8, -1)

Euclid step2 first coefficient multiple = -8

Euclid step2 first coefficient difference := sub(1, -8)

Euclid step2 first coefficient difference = 9

Euclid step2 second coefficient multiple := mul(8, 6)

Euclid step2 second coefficient multiple = 48

Euclid step2 second coefficient difference := sub(-5, 48)

Euclid step2 second coefficient difference = -53

step=2;two original-absolute-input coefficient pairs=[[1, -5], [-1, 6]];same quotient=8

next remainder coefficient pair=[9, -53];abs(a)*s_next+abs(b)*t_next=3

Euclid step3 exact ratio := div(12, 3)

Euclid step3 exact ratio = 4

Euclid step3 quotient times divisor := mul(4, 3)

Euclid step3 quotient times divisor = 12

Euclid step3 remainder := sub(12, 12)

Euclid step3 remainder = 0

step=3;left=12;positive right=3;exact ratio=4;floor quotient=4

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

Euclid step3 first coefficient multiple := mul(4, 9)

Euclid step3 first coefficient multiple = 36

Euclid step3 first coefficient difference := sub(-1, 36)

Euclid step3 first coefficient difference = -37

Euclid step3 second coefficient multiple := mul(4, -53)

Euclid step3 second coefficient multiple = -212

Euclid step3 second coefficient difference := sub(6, -212)

Euclid step3 second coefficient difference = 218

step=3;two original-absolute-input coefficient pairs=[[-1, 6], [9, -53]];same quotient=4

next remainder coefficient pair=[-37, 218];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[654, 111, 5, 99], [111, 99, 1, 12], [99, 12, 8, 3], [12, 3, 4, 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, 9)

Euclid restore original a sign in coefficient = -9

Euclid restore original b sign in coefficient := mul(1, -53)

Euclid restore original b sign in coefficient = -53

Euclid ORIGINAL signed a times x := mul(-654, -9)

Euclid ORIGINAL signed a times x = 5886

Euclid ORIGINAL signed b times y := mul(111, -53)

Euclid ORIGINAL signed b times y = -5883

Euclid final signed Bezout linear combination := sum(5886, -5883)

Euclid final signed Bezout linear combination = 3

original a=-654,b=111;deterministic absolute-chain coefficients=[9, -53]

sign-restored source coefficients x=-9,y=-53;a*x+b*y=3=nonnegative gcd

逐步反向核對

• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• 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, -9, -53, 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。