第 18803 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":447,"b":93}。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":447,"b":93}。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=447,b=93

initial nonnegative chain values=[447, 93];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(447, 93)

Euclid step0 exact ratio = 149/31

Euclid step0 quotient times divisor := mul(4, 93)

Euclid step0 quotient times divisor = 372

Euclid step0 remainder := sub(447, 372)

Euclid step0 remainder = 75

step=0;left=447;positive right=93;exact ratio=149/31;floor quotient=4

left=right*q+r with r=75 in[0,93);gcd pair=[3, 3]

Euclid step0 first coefficient multiple := mul(4, 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(4, 1)

Euclid step0 second coefficient multiple = 4

Euclid step0 second coefficient difference := sub(0, 4)

Euclid step0 second coefficient difference = -4

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=4

next remainder coefficient pair=[1, -4];abs(a)*s_next+abs(b)*t_next=75

Euclid step1 exact ratio := div(93, 75)

Euclid step1 exact ratio = 31/25

Euclid step1 quotient times divisor := mul(1, 75)

Euclid step1 quotient times divisor = 75

Euclid step1 remainder := sub(93, 75)

Euclid step1 remainder = 18

step=1;left=93;positive right=75;exact ratio=31/25;floor quotient=1

left=right*q+r with r=18 in[0,75);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, -4)

Euclid step1 second coefficient multiple = -4

Euclid step1 second coefficient difference := sub(1, -4)

Euclid step1 second coefficient difference = 5

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -4]];same quotient=1

next remainder coefficient pair=[-1, 5];abs(a)*s_next+abs(b)*t_next=18

Euclid step2 exact ratio := div(75, 18)

Euclid step2 exact ratio = 25/6

Euclid step2 quotient times divisor := mul(4, 18)

Euclid step2 quotient times divisor = 72

Euclid step2 remainder := sub(75, 72)

Euclid step2 remainder = 3

step=2;left=75;positive right=18;exact ratio=25/6;floor quotient=4

left=right*q+r with r=3 in[0,18);gcd pair=[3, 3]

Euclid step2 first coefficient multiple := mul(4, -1)

Euclid step2 first coefficient multiple = -4

Euclid step2 first coefficient difference := sub(1, -4)

Euclid step2 first coefficient difference = 5

Euclid step2 second coefficient multiple := mul(4, 5)

Euclid step2 second coefficient multiple = 20

Euclid step2 second coefficient difference := sub(-4, 20)

Euclid step2 second coefficient difference = -24

step=2;two original-absolute-input coefficient pairs=[[1, -4], [-1, 5]];same quotient=4

next remainder coefficient pair=[5, -24];abs(a)*s_next+abs(b)*t_next=3

Euclid step3 exact ratio := div(18, 3)

Euclid step3 exact ratio = 6

Euclid step3 quotient times divisor := mul(6, 3)

Euclid step3 quotient times divisor = 18

Euclid step3 remainder := sub(18, 18)

Euclid step3 remainder = 0

step=3;left=18;positive right=3;exact ratio=6;floor quotient=6

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

Euclid step3 first coefficient multiple := mul(6, 5)

Euclid step3 first coefficient multiple = 30

Euclid step3 first coefficient difference := sub(-1, 30)

Euclid step3 first coefficient difference = -31

Euclid step3 second coefficient multiple := mul(6, -24)

Euclid step3 second coefficient multiple = -144

Euclid step3 second coefficient difference := sub(5, -144)

Euclid step3 second coefficient difference = 149

step=3;two original-absolute-input coefficient pairs=[[-1, 5], [5, -24]];same quotient=6

next remainder coefficient pair=[-31, 149];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[447, 93, 4, 75], [93, 75, 1, 18], [75, 18, 4, 3], [18, 3, 6, 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, 5)

Euclid restore original a sign in coefficient = 5

Euclid restore original b sign in coefficient := mul(1, -24)

Euclid restore original b sign in coefficient = -24

Euclid ORIGINAL signed a times x := mul(447, 5)

Euclid ORIGINAL signed a times x = 2235

Euclid ORIGINAL signed b times y := mul(93, -24)

Euclid ORIGINAL signed b times y = -2232

Euclid final signed Bezout linear combination := sum(2235, -2232)

Euclid final signed Bezout linear combination = 3

original a=447,b=93;deterministic absolute-chain coefficients=[5, -24]

sign-restored source coefficients x=5,y=-24;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, 5, -24, 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。