第 18013 題：中國剩餘定理

原題與已知條件

正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={"moduli":[28,42],"residues":[1,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={"moduli":[28,42],"residues":[1,15]}。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-chinese-remainder-theorem

逐步推導

source operation=crt-noncoprime-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary]

original residue slot0 exact quotient := div(1, 28)

original residue slot0 exact quotient = 1/28

original residue slot0 floor quotient multiple := mul(0, 28)

original residue slot0 floor quotient multiple = 0

original residue slot0 normalized remainder := sub(1, 0)

original residue slot0 normalized remainder = 1

original residue slot0:integer value=1;positive modulus=28;exact ratio=1/28;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,28)

original residue slot1 exact quotient := div(15, 42)

original residue slot1 exact quotient = 5/14

original residue slot1 floor quotient multiple := mul(0, 42)

original residue slot1 floor quotient multiple = 0

original residue slot1 normalized remainder := sub(15, 0)

original residue slot1 normalized remainder = 15

original residue slot1:integer value=15;positive modulus=42;exact ratio=5/14;floor quotient=0

value=q*modulus+remainder;normalized residue=15 in[0,42)

CRT modulus product prefix0 := mul(1, 28)

CRT modulus product prefix0 = 28

LCM prefix0 gcd:positive original gcd inputs=[1, 28]

Euclidean rows[a,b,q,r] with a=q*b+r=[[1, 28, 0, 1], [28, 1, 28, 0]];positive gcd=1

CRT LCM prefix0 quotient := div(1, 1)

CRT LCM prefix0 quotient = 1

CRT LCM prefix0 value := mul(1, 28)

CRT LCM prefix0 value = 28

CRT modulus product prefix1 := mul(28, 42)

CRT modulus product prefix1 = 1176

LCM prefix1 gcd:positive original gcd inputs=[28, 42]

Euclidean rows[a,b,q,r] with a=q*b+r=[[28, 42, 0, 28], [42, 28, 1, 14], [28, 14, 2, 0]];positive gcd=14

CRT LCM prefix1 quotient := div(28, 14)

CRT LCM prefix1 quotient = 2

CRT LCM prefix1 value := mul(2, 42)

CRT LCM prefix1 value = 84

original modulus pair0,1:positive original gcd inputs=[28, 42]

Euclidean rows[a,b,q,r] with a=q*b+r=[[28, 42, 0, 28], [42, 28, 1, 14], [28, 14, 2, 0]];positive gcd=14

CRT original pair0,1 residue difference := sub(1, 15)

CRT original pair0,1 residue difference = -14

CRT original pair0,1 difference divided by gcd := div(-14, 14)

CRT original pair0,1 difference divided by gcd = -1

original constraint slots=[0, 1];moduli=[28, 42];gcd=14;normalized residue difference=-14

difference/gcd=-1;pair coprime=否;pair compatible=是

original aligned moduli=[28, 42];original signed residues=[1, 15];normalized residues=[1, 15];all pair audits=[[0, 1, 14, 是]]

pairwise coprime=否;gcd compatible=是;product=1176;LCM period=84

merge original slot0 Euclidean step0 quotient fraction := div(1, 28)

merge original slot0 Euclidean step0 quotient fraction = 1/28

merge original slot0 Euclidean step0 divisor multiple := mul(0, 28)

merge original slot0 Euclidean step0 divisor multiple = 0

merge original slot0 Euclidean step0 remainder := sub(1, 0)

merge original slot0 Euclidean step0 remainder = 1

merge original slot0 Euclidean step0 first coefficient multiple := mul(0, 0)

merge original slot0 Euclidean step0 first coefficient multiple = 0

merge original slot0 Euclidean step0 first coefficient difference := sub(1, 0)

merge original slot0 Euclidean step0 first coefficient difference = 1

merge original slot0 Euclidean step0 second coefficient multiple := mul(0, 1)

merge original slot0 Euclidean step0 second coefficient multiple = 0

merge original slot0 Euclidean step0 second coefficient difference := sub(0, 0)

merge original slot0 Euclidean step0 second coefficient difference = 0

merge original slot0:step=0;current remainders=[1, 28];coefficient pairs=[[1, 0], [0, 1]];exact ratio=1/28;floor q=0

next remainder=1;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

merge original slot0 Euclidean step1 quotient fraction := div(28, 1)

merge original slot0 Euclidean step1 quotient fraction = 28

merge original slot0 Euclidean step1 divisor multiple := mul(28, 1)

merge original slot0 Euclidean step1 divisor multiple = 28

merge original slot0 Euclidean step1 remainder := sub(28, 28)

merge original slot0 Euclidean step1 remainder = 0

merge original slot0 Euclidean step1 first coefficient multiple := mul(28, 1)

merge original slot0 Euclidean step1 first coefficient multiple = 28

merge original slot0 Euclidean step1 first coefficient difference := sub(0, 28)

merge original slot0 Euclidean step1 first coefficient difference = -28

merge original slot0 Euclidean step1 second coefficient multiple := mul(28, 0)

merge original slot0 Euclidean step1 second coefficient multiple = 0

merge original slot0 Euclidean step1 second coefficient difference := sub(1, 0)

merge original slot0 Euclidean step1 second coefficient difference = 1

merge original slot0:step=1;current remainders=[28, 1];coefficient pairs=[[0, 1], [1, 0]];exact ratio=28;floor q=28

next remainder=0;next signed coefficient pair=[-28, 1];a*x+b*y equals next remainder

merge original slot0 final first Bezout term := mul(1, 1)

merge original slot0 final first Bezout term = 1

merge original slot0 final second Bezout term := mul(28, 0)

merge original slot0 final second Bezout term = 0

merge original slot0 final Bezout identity := sum(1, 0)

merge original slot0 final Bezout identity = 1

merge slot0 new residue minus old class := sub(1, 0)

merge slot0 new residue minus old class = 1

merge slot0 difference divided by gcd := div(1, 1)

merge slot0 difference divided by gcd = 1

merge slot0 reduced old modulus := div(1, 1)

merge slot0 reduced old modulus = 1

merge slot0 reduced new modulus := div(28, 1)

merge slot0 reduced new modulus = 28

merge slot0 new LCM period := mul(1, 28)

merge slot0 new LCM period = 28

merge slot0 reduced-unit inverse exact quotient := div(1, 28)

merge slot0 reduced-unit inverse exact quotient = 1/28

merge slot0 reduced-unit inverse floor quotient multiple := mul(0, 28)

merge slot0 reduced-unit inverse floor quotient multiple = 0

merge slot0 reduced-unit inverse normalized remainder := sub(1, 0)

merge slot0 reduced-unit inverse normalized remainder = 1

merge slot0 reduced-unit inverse:integer value=1;positive modulus=28;exact ratio=1/28;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,28)

merge slot0 parameter congruence product := mul(1, 1)

merge slot0 parameter congruence product = 1

merge slot0 normalized parameter exact quotient := div(1, 28)

merge slot0 normalized parameter exact quotient = 1/28

merge slot0 normalized parameter floor quotient multiple := mul(0, 28)

merge slot0 normalized parameter floor quotient multiple = 0

merge slot0 normalized parameter normalized remainder := sub(1, 0)

merge slot0 normalized parameter normalized remainder = 1

merge slot0 normalized parameter:integer value=1;positive modulus=28;exact ratio=1/28;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,28)

merge slot0 old modulus times parameter := mul(1, 1)

merge slot0 old modulus times parameter = 1

merge slot0 unnormalized common solution := sum(0, 1)

merge slot0 unnormalized common solution = 1

merge slot0 least class representative exact quotient := div(1, 28)

merge slot0 least class representative exact quotient = 1/28

merge slot0 least class representative floor quotient multiple := mul(0, 28)

merge slot0 least class representative floor quotient multiple = 0

merge slot0 least class representative normalized remainder := sub(1, 0)

merge slot0 least class representative normalized remainder = 1

merge slot0 least class representative:integer value=1;positive modulus=28;exact ratio=1/28;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,28)

old FULL solution class=0 mod1;new original constraint=1 mod28;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 28]

write old x=a+M*t;reduced equation gives t=1 mod28;ALL merged solutions x=1 mod28

merge original slot1 Euclidean step0 quotient fraction := div(28, 42)

merge original slot1 Euclidean step0 quotient fraction = 2/3

merge original slot1 Euclidean step0 divisor multiple := mul(0, 42)

merge original slot1 Euclidean step0 divisor multiple = 0

merge original slot1 Euclidean step0 remainder := sub(28, 0)

merge original slot1 Euclidean step0 remainder = 28

merge original slot1 Euclidean step0 first coefficient multiple := mul(0, 0)

merge original slot1 Euclidean step0 first coefficient multiple = 0

merge original slot1 Euclidean step0 first coefficient difference := sub(1, 0)

merge original slot1 Euclidean step0 first coefficient difference = 1

merge original slot1 Euclidean step0 second coefficient multiple := mul(0, 1)

merge original slot1 Euclidean step0 second coefficient multiple = 0

merge original slot1 Euclidean step0 second coefficient difference := sub(0, 0)

merge original slot1 Euclidean step0 second coefficient difference = 0

merge original slot1:step=0;current remainders=[28, 42];coefficient pairs=[[1, 0], [0, 1]];exact ratio=2/3;floor q=0

next remainder=28;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

merge original slot1 Euclidean step1 quotient fraction := div(42, 28)

merge original slot1 Euclidean step1 quotient fraction = 3/2

merge original slot1 Euclidean step1 divisor multiple := mul(1, 28)

merge original slot1 Euclidean step1 divisor multiple = 28

merge original slot1 Euclidean step1 remainder := sub(42, 28)

merge original slot1 Euclidean step1 remainder = 14

merge original slot1 Euclidean step1 first coefficient multiple := mul(1, 1)

merge original slot1 Euclidean step1 first coefficient multiple = 1

merge original slot1 Euclidean step1 first coefficient difference := sub(0, 1)

merge original slot1 Euclidean step1 first coefficient difference = -1

merge original slot1 Euclidean step1 second coefficient multiple := mul(1, 0)

merge original slot1 Euclidean step1 second coefficient multiple = 0

merge original slot1 Euclidean step1 second coefficient difference := sub(1, 0)

merge original slot1 Euclidean step1 second coefficient difference = 1

merge original slot1:step=1;current remainders=[42, 28];coefficient pairs=[[0, 1], [1, 0]];exact ratio=3/2;floor q=1

next remainder=14;next signed coefficient pair=[-1, 1];a*x+b*y equals next remainder

merge original slot1 Euclidean step2 quotient fraction := div(28, 14)

merge original slot1 Euclidean step2 quotient fraction = 2

merge original slot1 Euclidean step2 divisor multiple := mul(2, 14)

merge original slot1 Euclidean step2 divisor multiple = 28

merge original slot1 Euclidean step2 remainder := sub(28, 28)

merge original slot1 Euclidean step2 remainder = 0

merge original slot1 Euclidean step2 first coefficient multiple := mul(2, -1)

merge original slot1 Euclidean step2 first coefficient multiple = -2

merge original slot1 Euclidean step2 first coefficient difference := sub(1, -2)

merge original slot1 Euclidean step2 first coefficient difference = 3

merge original slot1 Euclidean step2 second coefficient multiple := mul(2, 1)

merge original slot1 Euclidean step2 second coefficient multiple = 2

merge original slot1 Euclidean step2 second coefficient difference := sub(0, 2)

merge original slot1 Euclidean step2 second coefficient difference = -2

merge original slot1:step=2;current remainders=[28, 14];coefficient pairs=[[1, 0], [-1, 1]];exact ratio=2;floor q=2

next remainder=0;next signed coefficient pair=[3, -2];a*x+b*y equals next remainder

merge original slot1 final first Bezout term := mul(28, -1)

merge original slot1 final first Bezout term = -28

merge original slot1 final second Bezout term := mul(42, 1)

merge original slot1 final second Bezout term = 42

merge original slot1 final Bezout identity := sum(-28, 42)

merge original slot1 final Bezout identity = 14

merge slot1 new residue minus old class := sub(15, 1)

merge slot1 new residue minus old class = 14

merge slot1 difference divided by gcd := div(14, 14)

merge slot1 difference divided by gcd = 1

merge slot1 reduced old modulus := div(28, 14)

merge slot1 reduced old modulus = 2

merge slot1 reduced new modulus := div(42, 14)

merge slot1 reduced new modulus = 3

merge slot1 new LCM period := mul(28, 3)

merge slot1 new LCM period = 84

merge slot1 reduced-unit inverse exact quotient := div(-1, 3)

merge slot1 reduced-unit inverse exact quotient = -1/3

merge slot1 reduced-unit inverse floor quotient multiple := mul(-1, 3)

merge slot1 reduced-unit inverse floor quotient multiple = -3

merge slot1 reduced-unit inverse normalized remainder := sub(-1, -3)

merge slot1 reduced-unit inverse normalized remainder = 2

merge slot1 reduced-unit inverse:integer value=-1;positive modulus=3;exact ratio=-1/3;floor quotient=-1

value=q*modulus+remainder;normalized residue=2 in[0,3)

merge slot1 parameter congruence product := mul(1, 2)

merge slot1 parameter congruence product = 2

merge slot1 normalized parameter exact quotient := div(2, 3)

merge slot1 normalized parameter exact quotient = 2/3

merge slot1 normalized parameter floor quotient multiple := mul(0, 3)

merge slot1 normalized parameter floor quotient multiple = 0

merge slot1 normalized parameter normalized remainder := sub(2, 0)

merge slot1 normalized parameter normalized remainder = 2

merge slot1 normalized parameter:integer value=2;positive modulus=3;exact ratio=2/3;floor quotient=0

value=q*modulus+remainder;normalized residue=2 in[0,3)

merge slot1 old modulus times parameter := mul(28, 2)

merge slot1 old modulus times parameter = 56

merge slot1 unnormalized common solution := sum(1, 56)

merge slot1 unnormalized common solution = 57

merge slot1 least class representative exact quotient := div(57, 84)

merge slot1 least class representative exact quotient = 19/28

merge slot1 least class representative floor quotient multiple := mul(0, 84)

merge slot1 least class representative floor quotient multiple = 0

merge slot1 least class representative normalized remainder := sub(57, 0)

merge slot1 least class representative normalized remainder = 57

merge slot1 least class representative:integer value=57;positive modulus=84;exact ratio=19/28;floor quotient=0

value=q*modulus+remainder;normalized residue=57 in[0,84)

old FULL solution class=1 mod28;new original constraint=15 mod42;gcd=14;Bezout coefficients=[-1, 1];reduced coprime moduli=[2, 3]

write old x=a+M*t;reduced equation gives t=2 mod3;ALL merged solutions x=57 mod84

逐步反向核對

• S110 · 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。
• S109 · 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。
• S108 · premise_and_result_audit — 以精確有理數重算 merge slot1 least class representative normalized remainder，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 merge slot1 least class representative floor quotient multiple，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 merge slot1 least class representative exact quotient，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 merge slot1 unnormalized common solution，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 merge slot1 old modulus times parameter，不以小數近似替代。
• S103 · 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。
• S102 · premise_and_result_audit — 以精確有理數重算 merge slot1 normalized parameter normalized remainder，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 merge slot1 normalized parameter floor quotient multiple，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 merge slot1 normalized parameter exact quotient，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 merge slot1 parameter congruence product，不以小數近似替代。
• S098 · 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。
• S097 · premise_and_result_audit — 以精確有理數重算 merge slot1 reduced-unit inverse normalized remainder，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 merge slot1 reduced-unit inverse floor quotient multiple，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 merge slot1 reduced-unit inverse exact quotient，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 merge slot1 new LCM period，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 merge slot1 reduced new modulus，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 merge slot1 reduced old modulus，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 merge slot1 difference divided by gcd，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 merge slot1 new residue minus old class，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 merge original slot1 final Bezout identity，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 merge original slot1 final second Bezout term，不以小數近似替代。
• S087 · premise_and_result_audit — 以精確有理數重算 merge original slot1 final first Bezout term，不以小數近似替代。
• S086 · 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。
• S085 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 second coefficient difference，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 second coefficient multiple，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 first coefficient difference，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 first coefficient multiple，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 remainder，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 divisor multiple，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step2 quotient fraction，不以小數近似替代。
• S078 · 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。
• S077 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 second coefficient difference，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 second coefficient multiple，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 first coefficient difference，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 first coefficient multiple，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 remainder，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 divisor multiple，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step1 quotient fraction，不以小數近似替代。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 second coefficient difference，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 second coefficient multiple，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 first coefficient difference，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 first coefficient multiple，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 remainder，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 divisor multiple，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step0 quotient fraction，不以小數近似替代。
• S062 · 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。
• S061 · 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。
• S060 · premise_and_result_audit — 以精確有理數重算 merge slot0 least class representative normalized remainder，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 merge slot0 least class representative floor quotient multiple，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 merge slot0 least class representative exact quotient，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 merge slot0 unnormalized common solution，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 merge slot0 old modulus times parameter，不以小數近似替代。
• S055 · 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。
• S054 · premise_and_result_audit — 以精確有理數重算 merge slot0 normalized parameter normalized remainder，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 merge slot0 normalized parameter floor quotient multiple，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 merge slot0 normalized parameter exact quotient，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 merge slot0 parameter congruence product，不以小數近似替代。
• S050 · 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。
• S049 · premise_and_result_audit — 以精確有理數重算 merge slot0 reduced-unit inverse normalized remainder，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 merge slot0 reduced-unit inverse floor quotient multiple，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 merge slot0 reduced-unit inverse exact quotient，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 merge slot0 new LCM period，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 merge slot0 reduced new modulus，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 merge slot0 reduced old modulus，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 merge slot0 difference divided by gcd，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 merge slot0 new residue minus old class，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 merge original slot0 final Bezout identity，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 merge original slot0 final second Bezout term，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 merge original slot0 final first Bezout term，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 merge original slot0 Euclidean step1 second coefficient difference，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 second coefficient multiple，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 first coefficient difference，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 first coefficient multiple，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 remainder，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 divisor multiple，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step1 quotient fraction，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 — 以精確有理數重算 merge original slot0 Euclidean step0 second coefficient difference，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 second coefficient multiple，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 first coefficient difference，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 first coefficient multiple，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 remainder，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 divisor multiple，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 merge original slot0 Euclidean step0 quotient fraction，不以小數近似替代。
• S022 · 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。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 CRT original pair0,1 difference divided by gcd，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 CRT original pair0,1 residue difference，不以小數近似替代。
• S018 · 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。
• S017 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix1 value，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix1 quotient，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 CRT modulus product prefix1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix0 value，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix0 quotient，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 CRT modulus product prefix0，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 original residue slot1 normalized remainder，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 original residue slot1 floor quotient multiple，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 original residue slot1 exact quotient，不以小數近似替代。
• S005 · 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。
• S004 · premise_and_result_audit — 以精確有理數重算 original residue slot0 normalized remainder，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 original residue slot0 floor quotient multiple，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 original residue slot0 exact quotient，不以小數近似替代。
• 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

([28, 42], [1, 15], 是, 57, 84, noncoprime_not_automatically_inconsistent)

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。