第 18824 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":-930,"b":135}。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":-930,"b":135}。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=-930,b=135

initial nonnegative chain values=[930, 135];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(930, 135)

Euclid step0 exact ratio = 62/9

Euclid step0 quotient times divisor := mul(6, 135)

Euclid step0 quotient times divisor = 810

Euclid step0 remainder := sub(930, 810)

Euclid step0 remainder = 120

step=0;left=930;positive right=135;exact ratio=62/9;floor quotient=6

left=right*q+r with r=120 in[0,135);gcd pair=[15, 15]

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=120

Euclid step1 exact ratio := div(135, 120)

Euclid step1 exact ratio = 9/8

Euclid step1 quotient times divisor := mul(1, 120)

Euclid step1 quotient times divisor = 120

Euclid step1 remainder := sub(135, 120)

Euclid step1 remainder = 15

step=1;left=135;positive right=120;exact ratio=9/8;floor quotient=1

left=right*q+r with r=15 in[0,120);gcd pair=[15, 15]

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, -6)

Euclid step1 second coefficient multiple = -6

Euclid step1 second coefficient difference := sub(1, -6)

Euclid step1 second coefficient difference = 7

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

next remainder coefficient pair=[-1, 7];abs(a)*s_next+abs(b)*t_next=15

Euclid step2 exact ratio := div(120, 15)

Euclid step2 exact ratio = 8

Euclid step2 quotient times divisor := mul(8, 15)

Euclid step2 quotient times divisor = 120

Euclid step2 remainder := sub(120, 120)

Euclid step2 remainder = 0

step=2;left=120;positive right=15;exact ratio=8;floor quotient=8

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

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, 7)

Euclid step2 second coefficient multiple = 56

Euclid step2 second coefficient difference := sub(-6, 56)

Euclid step2 second coefficient difference = -62

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-1, 7]];same quotient=8

next remainder coefficient pair=[9, -62];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[930, 135, 6, 120], [135, 120, 1, 15], [120, 15, 8, 0]];terminal pair=[15, 0]

gcd(a,b)=gcd(15,0)=15;all divisions used positive divisors;nonnegative gcd convention

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

Euclid restore original a sign in coefficient = 1

Euclid restore original b sign in coefficient := mul(1, 7)

Euclid restore original b sign in coefficient = 7

Euclid ORIGINAL signed a times x := mul(-930, 1)

Euclid ORIGINAL signed a times x = -930

Euclid ORIGINAL signed b times y := mul(135, 7)

Euclid ORIGINAL signed b times y = 945

Euclid final signed Bezout linear combination := sum(-930, 945)

Euclid final signed Bezout linear combination = 15

original a=-930,b=135;deterministic absolute-chain coefficients=[-1, 7]

sign-restored source coefficients x=1,y=7;a*x+b*y=15=nonnegative gcd

逐步反向核對

• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• 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

(15, 1, 7, 15, 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。