第 18077 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-uniqueness-mod-product；資料={"moduli":[3,7],"residues":[1,3],"other":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 正moduli、normalized residues與exact integer congruences。操作=crt-uniqueness-mod-product；資料={"moduli":[3,7],"residues":[1,3],"other":31}。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-uniqueness-mod-product;original snapshot and mathematics retained

explicit ordered source output fields=[solution, other, product, same_residue_class, boundary]

original residue slot0 exact quotient := div(1, 3)

original residue slot0 exact quotient = 1/3

original residue slot0 floor quotient multiple := mul(0, 3)

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=3;exact ratio=1/3;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,3)

original residue slot1 exact quotient := div(3, 7)

original residue slot1 exact quotient = 3/7

original residue slot1 floor quotient multiple := mul(0, 7)

original residue slot1 floor quotient multiple = 0

original residue slot1 normalized remainder := sub(3, 0)

original residue slot1 normalized remainder = 3

original residue slot1:integer value=3;positive modulus=7;exact ratio=3/7;floor quotient=0

value=q*modulus+remainder;normalized residue=3 in[0,7)

CRT modulus product prefix0 := mul(1, 3)

CRT modulus product prefix0 = 3

LCM prefix0 gcd:positive original gcd inputs=[1, 3]

Euclidean rows[a,b,q,r] with a=q*b+r=[[1, 3, 0, 1], [3, 1, 3, 0]];positive gcd=1

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

CRT LCM prefix0 quotient = 1

CRT LCM prefix0 value := mul(1, 3)

CRT LCM prefix0 value = 3

CRT modulus product prefix1 := mul(3, 7)

CRT modulus product prefix1 = 21

LCM prefix1 gcd:positive original gcd inputs=[3, 7]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 7, 0, 3], [7, 3, 2, 1], [3, 1, 3, 0]];positive gcd=1

CRT LCM prefix1 quotient := div(3, 1)

CRT LCM prefix1 quotient = 3

CRT LCM prefix1 value := mul(3, 7)

CRT LCM prefix1 value = 21

original modulus pair0,1:positive original gcd inputs=[3, 7]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 7, 0, 3], [7, 3, 2, 1], [3, 1, 3, 0]];positive gcd=1

CRT original pair0,1 residue difference := sub(1, 3)

CRT original pair0,1 residue difference = -2

CRT original pair0,1 difference divided by gcd := div(-2, 1)

CRT original pair0,1 difference divided by gcd = -2

original constraint slots=[0, 1];moduli=[3, 7];gcd=1;normalized residue difference=-2

difference/gcd=-2;pair coprime=是;pair compatible=是

original aligned moduli=[3, 7];original signed residues=[1, 3];normalized residues=[1, 3];all pair audits=[[0, 1, 1, 是]]

pairwise coprime=是;gcd compatible=是;product=21;LCM period=21

merge original slot0 Euclidean step0 quotient fraction := div(1, 3)

merge original slot0 Euclidean step0 quotient fraction = 1/3

merge original slot0 Euclidean step0 divisor multiple := mul(0, 3)

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, 3];coefficient pairs=[[1, 0], [0, 1]];exact ratio=1/3;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(3, 1)

merge original slot0 Euclidean step1 quotient fraction = 3

merge original slot0 Euclidean step1 divisor multiple := mul(3, 1)

merge original slot0 Euclidean step1 divisor multiple = 3

merge original slot0 Euclidean step1 remainder := sub(3, 3)

merge original slot0 Euclidean step1 remainder = 0

merge original slot0 Euclidean step1 first coefficient multiple := mul(3, 1)

merge original slot0 Euclidean step1 first coefficient multiple = 3

merge original slot0 Euclidean step1 first coefficient difference := sub(0, 3)

merge original slot0 Euclidean step1 first coefficient difference = -3

merge original slot0 Euclidean step1 second coefficient multiple := mul(3, 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=[3, 1];coefficient pairs=[[0, 1], [1, 0]];exact ratio=3;floor q=3

next remainder=0;next signed coefficient pair=[-3, 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(3, 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(3, 1)

merge slot0 reduced new modulus = 3

merge slot0 new LCM period := mul(1, 3)

merge slot0 new LCM period = 3

merge slot0 reduced-unit inverse exact quotient := div(1, 3)

merge slot0 reduced-unit inverse exact quotient = 1/3

merge slot0 reduced-unit inverse floor quotient multiple := mul(0, 3)

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=3;exact ratio=1/3;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,3)

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

merge slot0 parameter congruence product = 1

merge slot0 normalized parameter exact quotient := div(1, 3)

merge slot0 normalized parameter exact quotient = 1/3

merge slot0 normalized parameter floor quotient multiple := mul(0, 3)

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=3;exact ratio=1/3;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,3)

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, 3)

merge slot0 least class representative exact quotient = 1/3

merge slot0 least class representative floor quotient multiple := mul(0, 3)

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=3;exact ratio=1/3;floor quotient=0

value=q*modulus+remainder;normalized residue=1 in[0,3)

old FULL solution class=0 mod1;new original constraint=1 mod3;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 3]

write old x=a+M*t;reduced equation gives t=1 mod3;ALL merged solutions x=1 mod3

merge original slot1 Euclidean step0 quotient fraction := div(3, 7)

merge original slot1 Euclidean step0 quotient fraction = 3/7

merge original slot1 Euclidean step0 divisor multiple := mul(0, 7)

merge original slot1 Euclidean step0 divisor multiple = 0

merge original slot1 Euclidean step0 remainder := sub(3, 0)

merge original slot1 Euclidean step0 remainder = 3

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=[3, 7];coefficient pairs=[[1, 0], [0, 1]];exact ratio=3/7;floor q=0

next remainder=3;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

merge original slot1 Euclidean step1 quotient fraction := div(7, 3)

merge original slot1 Euclidean step1 quotient fraction = 7/3

merge original slot1 Euclidean step1 divisor multiple := mul(2, 3)

merge original slot1 Euclidean step1 divisor multiple = 6

merge original slot1 Euclidean step1 remainder := sub(7, 6)

merge original slot1 Euclidean step1 remainder = 1

merge original slot1 Euclidean step1 first coefficient multiple := mul(2, 1)

merge original slot1 Euclidean step1 first coefficient multiple = 2

merge original slot1 Euclidean step1 first coefficient difference := sub(0, 2)

merge original slot1 Euclidean step1 first coefficient difference = -2

merge original slot1 Euclidean step1 second coefficient multiple := mul(2, 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=[7, 3];coefficient pairs=[[0, 1], [1, 0]];exact ratio=7/3;floor q=2

next remainder=1;next signed coefficient pair=[-2, 1];a*x+b*y equals next remainder

merge original slot1 Euclidean step2 quotient fraction := div(3, 1)

merge original slot1 Euclidean step2 quotient fraction = 3

merge original slot1 Euclidean step2 divisor multiple := mul(3, 1)

merge original slot1 Euclidean step2 divisor multiple = 3

merge original slot1 Euclidean step2 remainder := sub(3, 3)

merge original slot1 Euclidean step2 remainder = 0

merge original slot1 Euclidean step2 first coefficient multiple := mul(3, -2)

merge original slot1 Euclidean step2 first coefficient multiple = -6

merge original slot1 Euclidean step2 first coefficient difference := sub(1, -6)

merge original slot1 Euclidean step2 first coefficient difference = 7

merge original slot1 Euclidean step2 second coefficient multiple := mul(3, 1)

merge original slot1 Euclidean step2 second coefficient multiple = 3

merge original slot1 Euclidean step2 second coefficient difference := sub(0, 3)

merge original slot1 Euclidean step2 second coefficient difference = -3

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

next remainder=0;next signed coefficient pair=[7, -3];a*x+b*y equals next remainder

merge original slot1 final first Bezout term := mul(3, -2)

merge original slot1 final first Bezout term = -6

merge original slot1 final second Bezout term := mul(7, 1)

merge original slot1 final second Bezout term = 7

merge original slot1 final Bezout identity := sum(-6, 7)

merge original slot1 final Bezout identity = 1

merge slot1 new residue minus old class := sub(3, 1)

merge slot1 new residue minus old class = 2

merge slot1 difference divided by gcd := div(2, 1)

merge slot1 difference divided by gcd = 2

merge slot1 reduced old modulus := div(3, 1)

merge slot1 reduced old modulus = 3

merge slot1 reduced new modulus := div(7, 1)

merge slot1 reduced new modulus = 7

merge slot1 new LCM period := mul(3, 7)

merge slot1 new LCM period = 21

merge slot1 reduced-unit inverse exact quotient := div(-2, 7)

merge slot1 reduced-unit inverse exact quotient = -2/7

merge slot1 reduced-unit inverse floor quotient multiple := mul(-1, 7)

merge slot1 reduced-unit inverse floor quotient multiple = -7

merge slot1 reduced-unit inverse normalized remainder := sub(-2, -7)

merge slot1 reduced-unit inverse normalized remainder = 5

merge slot1 reduced-unit inverse:integer value=-2;positive modulus=7;exact ratio=-2/7;floor quotient=-1

value=q*modulus+remainder;normalized residue=5 in[0,7)

merge slot1 parameter congruence product := mul(2, 5)

merge slot1 parameter congruence product = 10

merge slot1 normalized parameter exact quotient := div(10, 7)

merge slot1 normalized parameter exact quotient = 10/7

merge slot1 normalized parameter floor quotient multiple := mul(1, 7)

merge slot1 normalized parameter floor quotient multiple = 7

merge slot1 normalized parameter normalized remainder := sub(10, 7)

merge slot1 normalized parameter normalized remainder = 3

merge slot1 normalized parameter:integer value=10;positive modulus=7;exact ratio=10/7;floor quotient=1

value=q*modulus+remainder;normalized residue=3 in[0,7)

merge slot1 old modulus times parameter := mul(3, 3)

merge slot1 old modulus times parameter = 9

merge slot1 unnormalized common solution := sum(1, 9)

merge slot1 unnormalized common solution = 10

merge slot1 least class representative exact quotient := div(10, 21)

merge slot1 least class representative exact quotient = 10/21

merge slot1 least class representative floor quotient multiple := mul(0, 21)

merge slot1 least class representative floor quotient multiple = 0

merge slot1 least class representative normalized remainder := sub(10, 0)

merge slot1 least class representative normalized remainder = 10

merge slot1 least class representative:integer value=10;positive modulus=21;exact ratio=10/21;floor quotient=0

value=q*modulus+remainder;normalized residue=10 in[0,21)

old FULL solution class=1 mod3;new original constraint=3 mod7;gcd=1;Bezout coefficients=[-2, 1];reduced coprime moduli=[3, 7]

write old x=a+M*t;reduced equation gives t=3 mod7;ALL merged solutions x=10 mod21

CRT comparison integer minus least solution := sub(31, 10)

CRT comparison integer minus least solution = 21

CRT comparison solution-class test exact quotient := div(21, 21)

CRT comparison solution-class test exact quotient = 1

CRT comparison solution-class test floor quotient multiple := mul(1, 21)

CRT comparison solution-class test floor quotient multiple = 21

CRT comparison solution-class test normalized remainder := sub(21, 21)

CRT comparison solution-class test normalized remainder = 0

CRT comparison solution-class test:integer value=21;positive modulus=21;exact ratio=1;floor quotient=1

value=q*modulus+remainder;normalized residue=0 in[0,21)

least solution=10;arbitrary comparison integer=31;product period=21;difference=21

same residue class=是;unique_modulo_product_not_unique_integer

逐步反向核對

• S116 · 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。
• S115 · 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。
• S114 · premise_and_result_audit — 以精確有理數重算 CRT comparison solution-class test normalized remainder，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 CRT comparison solution-class test floor quotient multiple，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 CRT comparison solution-class test exact quotient，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 CRT comparison integer minus least solution，不以小數近似替代。
• 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

(10, 31, 21, 是, unique_modulo_product_not_unique_integer)

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。