第 18060 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-solve-system；資料={"moduli":[3,5,23],"residues":[9,19,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 正moduli、normalized residues與exact integer congruences。操作=crt-solve-system；資料={"moduli":[3,5,23],"residues":[9,19,29]}。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-solve-system;original snapshot and mathematics retained

explicit ordered source output fields=[moduli, normalized_residues, solution, product, congruence_checks, solved]

original residue slot0 exact quotient := div(9, 3)

original residue slot0 exact quotient = 3

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

original residue slot0 floor quotient multiple = 9

original residue slot0 normalized remainder := sub(9, 9)

original residue slot0 normalized remainder = 0

original residue slot0:integer value=9;positive modulus=3;exact ratio=3;floor quotient=3

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

original residue slot1 exact quotient := div(19, 5)

original residue slot1 exact quotient = 19/5

original residue slot1 floor quotient multiple := mul(3, 5)

original residue slot1 floor quotient multiple = 15

original residue slot1 normalized remainder := sub(19, 15)

original residue slot1 normalized remainder = 4

original residue slot1:integer value=19;positive modulus=5;exact ratio=19/5;floor quotient=3

value=q*modulus+remainder;normalized residue=4 in[0,5)

original residue slot2 exact quotient := div(29, 23)

original residue slot2 exact quotient = 29/23

original residue slot2 floor quotient multiple := mul(1, 23)

original residue slot2 floor quotient multiple = 23

original residue slot2 normalized remainder := sub(29, 23)

original residue slot2 normalized remainder = 6

original residue slot2:integer value=29;positive modulus=23;exact ratio=29/23;floor quotient=1

value=q*modulus+remainder;normalized residue=6 in[0,23)

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, 5)

CRT modulus product prefix1 = 15

LCM prefix1 gcd:positive original gcd inputs=[3, 5]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 5, 0, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

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

CRT LCM prefix1 quotient = 3

CRT LCM prefix1 value := mul(3, 5)

CRT LCM prefix1 value = 15

CRT modulus product prefix2 := mul(15, 23)

CRT modulus product prefix2 = 345

LCM prefix2 gcd:positive original gcd inputs=[15, 23]

Euclidean rows[a,b,q,r] with a=q*b+r=[[15, 23, 0, 15], [23, 15, 1, 8], [15, 8, 1, 7], [8, 7, 1, 1], [7, 1, 7, 0]];positive gcd=1

CRT LCM prefix2 quotient := div(15, 1)

CRT LCM prefix2 quotient = 15

CRT LCM prefix2 value := mul(15, 23)

CRT LCM prefix2 value = 345

original modulus pair0,1:positive original gcd inputs=[3, 5]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 5, 0, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

CRT original pair0,1 residue difference := sub(0, 4)

CRT original pair0,1 residue difference = -4

CRT original pair0,1 difference divided by gcd := div(-4, 1)

CRT original pair0,1 difference divided by gcd = -4

original constraint slots=[0, 1];moduli=[3, 5];gcd=1;normalized residue difference=-4

difference/gcd=-4;pair coprime=是;pair compatible=是

original modulus pair0,2:positive original gcd inputs=[3, 23]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 23, 0, 3], [23, 3, 7, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

CRT original pair0,2 residue difference := sub(0, 6)

CRT original pair0,2 residue difference = -6

CRT original pair0,2 difference divided by gcd := div(-6, 1)

CRT original pair0,2 difference divided by gcd = -6

original constraint slots=[0, 2];moduli=[3, 23];gcd=1;normalized residue difference=-6

difference/gcd=-6;pair coprime=是;pair compatible=是

original modulus pair1,2:positive original gcd inputs=[5, 23]

Euclidean rows[a,b,q,r] with a=q*b+r=[[5, 23, 0, 5], [23, 5, 4, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

CRT original pair1,2 residue difference := sub(4, 6)

CRT original pair1,2 residue difference = -2

CRT original pair1,2 difference divided by gcd := div(-2, 1)

CRT original pair1,2 difference divided by gcd = -2

original constraint slots=[1, 2];moduli=[5, 23];gcd=1;normalized residue difference=-2

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

original aligned moduli=[3, 5, 23];original signed residues=[9, 19, 29];normalized residues=[0, 4, 6];all pair audits=[[0, 1, 1, 是], [0, 2, 1, 是], [1, 2, 1, 是]]

pairwise coprime=是;gcd compatible=是;product=345;LCM period=345

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(0, 0)

merge slot0 new residue minus old class = 0

merge slot0 difference divided by gcd := div(0, 1)

merge slot0 difference divided by gcd = 0

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(0, 1)

merge slot0 parameter congruence product = 0

merge slot0 normalized parameter exact quotient := div(0, 3)

merge slot0 normalized parameter exact quotient = 0

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(0, 0)

merge slot0 normalized parameter normalized remainder = 0

merge slot0 normalized parameter:integer value=0;positive modulus=3;exact ratio=0;floor quotient=0

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

merge slot0 old modulus times parameter := mul(1, 0)

merge slot0 old modulus times parameter = 0

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

merge slot0 unnormalized common solution = 0

merge slot0 least class representative exact quotient := div(0, 3)

merge slot0 least class representative exact quotient = 0

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(0, 0)

merge slot0 least class representative normalized remainder = 0

merge slot0 least class representative:integer value=0;positive modulus=3;exact ratio=0;floor quotient=0

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

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

write old x=a+M*t;reduced equation gives t=0 mod3;ALL merged solutions x=0 mod3

merge original slot1 Euclidean step0 quotient fraction := div(3, 5)

merge original slot1 Euclidean step0 quotient fraction = 3/5

merge original slot1 Euclidean step0 divisor multiple := mul(0, 5)

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, 5];coefficient pairs=[[1, 0], [0, 1]];exact ratio=3/5;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(5, 3)

merge original slot1 Euclidean step1 quotient fraction = 5/3

merge original slot1 Euclidean step1 divisor multiple := mul(1, 3)

merge original slot1 Euclidean step1 divisor multiple = 3

merge original slot1 Euclidean step1 remainder := sub(5, 3)

merge original slot1 Euclidean step1 remainder = 2

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=[5, 3];coefficient pairs=[[0, 1], [1, 0]];exact ratio=5/3;floor q=1

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

merge original slot1 Euclidean step2 quotient fraction := div(3, 2)

merge original slot1 Euclidean step2 quotient fraction = 3/2

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

merge original slot1 Euclidean step2 divisor multiple = 2

merge original slot1 Euclidean step2 remainder := sub(3, 2)

merge original slot1 Euclidean step2 remainder = 1

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

merge original slot1 Euclidean step2 first coefficient multiple = -1

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

merge original slot1 Euclidean step2 first coefficient difference = 2

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

merge original slot1 Euclidean step2 second coefficient multiple = 1

merge original slot1 Euclidean step2 second coefficient difference := sub(0, 1)

merge original slot1 Euclidean step2 second coefficient difference = -1

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

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

merge original slot1 Euclidean step3 quotient fraction := div(2, 1)

merge original slot1 Euclidean step3 quotient fraction = 2

merge original slot1 Euclidean step3 divisor multiple := mul(2, 1)

merge original slot1 Euclidean step3 divisor multiple = 2

merge original slot1 Euclidean step3 remainder := sub(2, 2)

merge original slot1 Euclidean step3 remainder = 0

merge original slot1 Euclidean step3 first coefficient multiple := mul(2, 2)

merge original slot1 Euclidean step3 first coefficient multiple = 4

merge original slot1 Euclidean step3 first coefficient difference := sub(-1, 4)

merge original slot1 Euclidean step3 first coefficient difference = -5

merge original slot1 Euclidean step3 second coefficient multiple := mul(2, -1)

merge original slot1 Euclidean step3 second coefficient multiple = -2

merge original slot1 Euclidean step3 second coefficient difference := sub(1, -2)

merge original slot1 Euclidean step3 second coefficient difference = 3

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

next remainder=0;next signed coefficient pair=[-5, 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(5, -1)

merge original slot1 final second Bezout term = -5

merge original slot1 final Bezout identity := sum(6, -5)

merge original slot1 final Bezout identity = 1

merge slot1 new residue minus old class := sub(4, 0)

merge slot1 new residue minus old class = 4

merge slot1 difference divided by gcd := div(4, 1)

merge slot1 difference divided by gcd = 4

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

merge slot1 reduced old modulus = 3

merge slot1 reduced new modulus := div(5, 1)

merge slot1 reduced new modulus = 5

merge slot1 new LCM period := mul(3, 5)

merge slot1 new LCM period = 15

merge slot1 reduced-unit inverse exact quotient := div(2, 5)

merge slot1 reduced-unit inverse exact quotient = 2/5

merge slot1 reduced-unit inverse floor quotient multiple := mul(0, 5)

merge slot1 reduced-unit inverse floor quotient multiple = 0

merge slot1 reduced-unit inverse normalized remainder := sub(2, 0)

merge slot1 reduced-unit inverse normalized remainder = 2

merge slot1 reduced-unit inverse:integer value=2;positive modulus=5;exact ratio=2/5;floor quotient=0

value=q*modulus+remainder;normalized residue=2 in[0,5)

merge slot1 parameter congruence product := mul(4, 2)

merge slot1 parameter congruence product = 8

merge slot1 normalized parameter exact quotient := div(8, 5)

merge slot1 normalized parameter exact quotient = 8/5

merge slot1 normalized parameter floor quotient multiple := mul(1, 5)

merge slot1 normalized parameter floor quotient multiple = 5

merge slot1 normalized parameter normalized remainder := sub(8, 5)

merge slot1 normalized parameter normalized remainder = 3

merge slot1 normalized parameter:integer value=8;positive modulus=5;exact ratio=8/5;floor quotient=1

value=q*modulus+remainder;normalized residue=3 in[0,5)

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

merge slot1 old modulus times parameter = 9

merge slot1 unnormalized common solution := sum(0, 9)

merge slot1 unnormalized common solution = 9

merge slot1 least class representative exact quotient := div(9, 15)

merge slot1 least class representative exact quotient = 3/5

merge slot1 least class representative floor quotient multiple := mul(0, 15)

merge slot1 least class representative floor quotient multiple = 0

merge slot1 least class representative normalized remainder := sub(9, 0)

merge slot1 least class representative normalized remainder = 9

merge slot1 least class representative:integer value=9;positive modulus=15;exact ratio=3/5;floor quotient=0

value=q*modulus+remainder;normalized residue=9 in[0,15)

old FULL solution class=0 mod3;new original constraint=4 mod5;gcd=1;Bezout coefficients=[2, -1];reduced coprime moduli=[3, 5]

write old x=a+M*t;reduced equation gives t=3 mod5;ALL merged solutions x=9 mod15

merge original slot2 Euclidean step0 quotient fraction := div(15, 23)

merge original slot2 Euclidean step0 quotient fraction = 15/23

merge original slot2 Euclidean step0 divisor multiple := mul(0, 23)

merge original slot2 Euclidean step0 divisor multiple = 0

merge original slot2 Euclidean step0 remainder := sub(15, 0)

merge original slot2 Euclidean step0 remainder = 15

merge original slot2 Euclidean step0 first coefficient multiple := mul(0, 0)

merge original slot2 Euclidean step0 first coefficient multiple = 0

merge original slot2 Euclidean step0 first coefficient difference := sub(1, 0)

merge original slot2 Euclidean step0 first coefficient difference = 1

merge original slot2 Euclidean step0 second coefficient multiple := mul(0, 1)

merge original slot2 Euclidean step0 second coefficient multiple = 0

merge original slot2 Euclidean step0 second coefficient difference := sub(0, 0)

merge original slot2 Euclidean step0 second coefficient difference = 0

merge original slot2:step=0;current remainders=[15, 23];coefficient pairs=[[1, 0], [0, 1]];exact ratio=15/23;floor q=0

next remainder=15;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

merge original slot2 Euclidean step1 quotient fraction := div(23, 15)

merge original slot2 Euclidean step1 quotient fraction = 23/15

merge original slot2 Euclidean step1 divisor multiple := mul(1, 15)

merge original slot2 Euclidean step1 divisor multiple = 15

merge original slot2 Euclidean step1 remainder := sub(23, 15)

merge original slot2 Euclidean step1 remainder = 8

merge original slot2 Euclidean step1 first coefficient multiple := mul(1, 1)

merge original slot2 Euclidean step1 first coefficient multiple = 1

merge original slot2 Euclidean step1 first coefficient difference := sub(0, 1)

merge original slot2 Euclidean step1 first coefficient difference = -1

merge original slot2 Euclidean step1 second coefficient multiple := mul(1, 0)

merge original slot2 Euclidean step1 second coefficient multiple = 0

merge original slot2 Euclidean step1 second coefficient difference := sub(1, 0)

merge original slot2 Euclidean step1 second coefficient difference = 1

merge original slot2:step=1;current remainders=[23, 15];coefficient pairs=[[0, 1], [1, 0]];exact ratio=23/15;floor q=1

next remainder=8;next signed coefficient pair=[-1, 1];a*x+b*y equals next remainder

merge original slot2 Euclidean step2 quotient fraction := div(15, 8)

merge original slot2 Euclidean step2 quotient fraction = 15/8

merge original slot2 Euclidean step2 divisor multiple := mul(1, 8)

merge original slot2 Euclidean step2 divisor multiple = 8

merge original slot2 Euclidean step2 remainder := sub(15, 8)

merge original slot2 Euclidean step2 remainder = 7

merge original slot2 Euclidean step2 first coefficient multiple := mul(1, -1)

merge original slot2 Euclidean step2 first coefficient multiple = -1

merge original slot2 Euclidean step2 first coefficient difference := sub(1, -1)

merge original slot2 Euclidean step2 first coefficient difference = 2

merge original slot2 Euclidean step2 second coefficient multiple := mul(1, 1)

merge original slot2 Euclidean step2 second coefficient multiple = 1

merge original slot2 Euclidean step2 second coefficient difference := sub(0, 1)

merge original slot2 Euclidean step2 second coefficient difference = -1

merge original slot2:step=2;current remainders=[15, 8];coefficient pairs=[[1, 0], [-1, 1]];exact ratio=15/8;floor q=1

next remainder=7;next signed coefficient pair=[2, -1];a*x+b*y equals next remainder

merge original slot2 Euclidean step3 quotient fraction := div(8, 7)

merge original slot2 Euclidean step3 quotient fraction = 8/7

merge original slot2 Euclidean step3 divisor multiple := mul(1, 7)

merge original slot2 Euclidean step3 divisor multiple = 7

merge original slot2 Euclidean step3 remainder := sub(8, 7)

merge original slot2 Euclidean step3 remainder = 1

merge original slot2 Euclidean step3 first coefficient multiple := mul(1, 2)

merge original slot2 Euclidean step3 first coefficient multiple = 2

merge original slot2 Euclidean step3 first coefficient difference := sub(-1, 2)

merge original slot2 Euclidean step3 first coefficient difference = -3

merge original slot2 Euclidean step3 second coefficient multiple := mul(1, -1)

merge original slot2 Euclidean step3 second coefficient multiple = -1

merge original slot2 Euclidean step3 second coefficient difference := sub(1, -1)

merge original slot2 Euclidean step3 second coefficient difference = 2

merge original slot2:step=3;current remainders=[8, 7];coefficient pairs=[[-1, 1], [2, -1]];exact ratio=8/7;floor q=1

next remainder=1;next signed coefficient pair=[-3, 2];a*x+b*y equals next remainder

merge original slot2 Euclidean step4 quotient fraction := div(7, 1)

merge original slot2 Euclidean step4 quotient fraction = 7

merge original slot2 Euclidean step4 divisor multiple := mul(7, 1)

merge original slot2 Euclidean step4 divisor multiple = 7

merge original slot2 Euclidean step4 remainder := sub(7, 7)

merge original slot2 Euclidean step4 remainder = 0

merge original slot2 Euclidean step4 first coefficient multiple := mul(7, -3)

merge original slot2 Euclidean step4 first coefficient multiple = -21

merge original slot2 Euclidean step4 first coefficient difference := sub(2, -21)

merge original slot2 Euclidean step4 first coefficient difference = 23

merge original slot2 Euclidean step4 second coefficient multiple := mul(7, 2)

merge original slot2 Euclidean step4 second coefficient multiple = 14

merge original slot2 Euclidean step4 second coefficient difference := sub(-1, 14)

merge original slot2 Euclidean step4 second coefficient difference = -15

merge original slot2:step=4;current remainders=[7, 1];coefficient pairs=[[2, -1], [-3, 2]];exact ratio=7;floor q=7

next remainder=0;next signed coefficient pair=[23, -15];a*x+b*y equals next remainder

merge original slot2 final first Bezout term := mul(15, -3)

merge original slot2 final first Bezout term = -45

merge original slot2 final second Bezout term := mul(23, 2)

merge original slot2 final second Bezout term = 46

merge original slot2 final Bezout identity := sum(-45, 46)

merge original slot2 final Bezout identity = 1

merge slot2 new residue minus old class := sub(6, 9)

merge slot2 new residue minus old class = -3

merge slot2 difference divided by gcd := div(-3, 1)

merge slot2 difference divided by gcd = -3

merge slot2 reduced old modulus := div(15, 1)

merge slot2 reduced old modulus = 15

merge slot2 reduced new modulus := div(23, 1)

merge slot2 reduced new modulus = 23

merge slot2 new LCM period := mul(15, 23)

merge slot2 new LCM period = 345

merge slot2 reduced-unit inverse exact quotient := div(-3, 23)

merge slot2 reduced-unit inverse exact quotient = -3/23

merge slot2 reduced-unit inverse floor quotient multiple := mul(-1, 23)

merge slot2 reduced-unit inverse floor quotient multiple = -23

merge slot2 reduced-unit inverse normalized remainder := sub(-3, -23)

merge slot2 reduced-unit inverse normalized remainder = 20

merge slot2 reduced-unit inverse:integer value=-3;positive modulus=23;exact ratio=-3/23;floor quotient=-1

value=q*modulus+remainder;normalized residue=20 in[0,23)

merge slot2 parameter congruence product := mul(-3, 20)

merge slot2 parameter congruence product = -60

merge slot2 normalized parameter exact quotient := div(-60, 23)

merge slot2 normalized parameter exact quotient = -60/23

merge slot2 normalized parameter floor quotient multiple := mul(-3, 23)

merge slot2 normalized parameter floor quotient multiple = -69

merge slot2 normalized parameter normalized remainder := sub(-60, -69)

merge slot2 normalized parameter normalized remainder = 9

merge slot2 normalized parameter:integer value=-60;positive modulus=23;exact ratio=-60/23;floor quotient=-3

value=q*modulus+remainder;normalized residue=9 in[0,23)

merge slot2 old modulus times parameter := mul(15, 9)

merge slot2 old modulus times parameter = 135

merge slot2 unnormalized common solution := sum(9, 135)

merge slot2 unnormalized common solution = 144

merge slot2 least class representative exact quotient := div(144, 345)

merge slot2 least class representative exact quotient = 48/115

merge slot2 least class representative floor quotient multiple := mul(0, 345)

merge slot2 least class representative floor quotient multiple = 0

merge slot2 least class representative normalized remainder := sub(144, 0)

merge slot2 least class representative normalized remainder = 144

merge slot2 least class representative:integer value=144;positive modulus=345;exact ratio=48/115;floor quotient=0

value=q*modulus+remainder;normalized residue=144 in[0,345)

old FULL solution class=9 mod15;new original constraint=6 mod23;gcd=1;Bezout coefficients=[-3, 2];reduced coprime moduli=[15, 23]

write old x=a+M*t;reduced equation gives t=9 mod23;ALL merged solutions x=144 mod345

逐步反向核對

• S198 · 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。
• S197 · 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。
• S196 · premise_and_result_audit — 以精確有理數重算 merge slot2 least class representative normalized remainder，不以小數近似替代。
• S195 · premise_and_result_audit — 以精確有理數重算 merge slot2 least class representative floor quotient multiple，不以小數近似替代。
• S194 · premise_and_result_audit — 以精確有理數重算 merge slot2 least class representative exact quotient，不以小數近似替代。
• S193 · premise_and_result_audit — 以精確有理數重算 merge slot2 unnormalized common solution，不以小數近似替代。
• S192 · premise_and_result_audit — 以精確有理數重算 merge slot2 old modulus times parameter，不以小數近似替代。
• S191 · 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。
• S190 · premise_and_result_audit — 以精確有理數重算 merge slot2 normalized parameter normalized remainder，不以小數近似替代。
• S189 · premise_and_result_audit — 以精確有理數重算 merge slot2 normalized parameter floor quotient multiple，不以小數近似替代。
• S188 · premise_and_result_audit — 以精確有理數重算 merge slot2 normalized parameter exact quotient，不以小數近似替代。
• S187 · premise_and_result_audit — 以精確有理數重算 merge slot2 parameter congruence product，不以小數近似替代。
• S186 · 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。
• S185 · premise_and_result_audit — 以精確有理數重算 merge slot2 reduced-unit inverse normalized remainder，不以小數近似替代。
• S184 · premise_and_result_audit — 以精確有理數重算 merge slot2 reduced-unit inverse floor quotient multiple，不以小數近似替代。
• S183 · premise_and_result_audit — 以精確有理數重算 merge slot2 reduced-unit inverse exact quotient，不以小數近似替代。
• S182 · premise_and_result_audit — 以精確有理數重算 merge slot2 new LCM period，不以小數近似替代。
• S181 · premise_and_result_audit — 以精確有理數重算 merge slot2 reduced new modulus，不以小數近似替代。
• S180 · premise_and_result_audit — 以精確有理數重算 merge slot2 reduced old modulus，不以小數近似替代。
• S179 · premise_and_result_audit — 以精確有理數重算 merge slot2 difference divided by gcd，不以小數近似替代。
• S178 · premise_and_result_audit — 以精確有理數重算 merge slot2 new residue minus old class，不以小數近似替代。
• S177 · premise_and_result_audit — 以精確有理數重算 merge original slot2 final Bezout identity，不以小數近似替代。
• S176 · premise_and_result_audit — 以精確有理數重算 merge original slot2 final second Bezout term，不以小數近似替代。
• S175 · premise_and_result_audit — 以精確有理數重算 merge original slot2 final first Bezout term，不以小數近似替代。
• S174 · 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。
• S173 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 second coefficient difference，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 second coefficient multiple，不以小數近似替代。
• S171 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 first coefficient difference，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 first coefficient multiple，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 remainder，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 divisor multiple，不以小數近似替代。
• S167 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step4 quotient fraction，不以小數近似替代。
• S166 · 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。
• S165 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 second coefficient difference，不以小數近似替代。
• S164 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 second coefficient multiple，不以小數近似替代。
• S163 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 first coefficient difference，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 first coefficient multiple，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 remainder，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 divisor multiple，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step3 quotient fraction，不以小數近似替代。
• S158 · 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。
• S157 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 second coefficient difference，不以小數近似替代。
• S156 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 second coefficient multiple，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 first coefficient difference，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 first coefficient multiple，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 remainder，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 divisor multiple，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step2 quotient fraction，不以小數近似替代。
• S150 · 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。
• S149 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 second coefficient difference，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 second coefficient multiple，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 first coefficient difference，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 first coefficient multiple，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 remainder，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 divisor multiple，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step1 quotient fraction，不以小數近似替代。
• S142 · 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。
• S141 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 second coefficient difference，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 second coefficient multiple，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 first coefficient difference，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 first coefficient multiple，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 remainder，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 divisor multiple，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 merge original slot2 Euclidean step0 quotient fraction，不以小數近似替代。
• S134 · 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。
• S133 · 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。
• S132 · 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。
• S131 · 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。
• S130 · 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。
• S129 · 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。
• S128 · 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。
• S127 · 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。
• S126 · 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。
• S125 · 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。
• S124 · 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。
• S123 · 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。
• S122 · 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。
• S121 · 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。
• S120 · 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。
• S119 · 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。
• S118 · 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。
• S117 · 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。
• S116 · 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。
• S115 · 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。
• S114 · 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。
• S113 · 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。
• S112 · 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。
• S111 · 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。
• 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 — 以精確有理數重算 merge original slot1 Euclidean step3 second coefficient difference，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 second coefficient multiple，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 first coefficient difference，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 first coefficient multiple，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 remainder，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 divisor multiple，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 merge original slot1 Euclidean step3 quotient fraction，不以小數近似替代。
• S102 · 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。
• S101 · 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。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• S096 · 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。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• 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 step0 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 step0 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 step0 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 step0 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 step0 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 step0 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 step0 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 CRT original pair1,2 difference divided by gcd，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 CRT original pair1,2 residue difference，不以小數近似替代。
• S034 · 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。
• 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 CRT original pair0,2 difference divided by gcd，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 CRT original pair0,2 residue difference，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 CRT original pair0,1 difference divided by gcd，不以小數近似替代。
• S027 · 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。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix2 value，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix2 quotient，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 CRT modulus product prefix2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix1 value，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix1 quotient，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 CRT modulus product prefix1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix0 value，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 CRT LCM prefix0 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 prefix0，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 original residue slot2 normalized remainder，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 original residue slot2 floor quotient multiple，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 original residue slot2 exact quotient，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

([3, 5, 23], [0, 4, 6], 144, 345, [是, 是, 是], 是)

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。