Question 18072: Every Congruence Must Pass in the Same CRT System

Original question and givens

Use the complete integer congruence system with positive distinct moduli=[3, 5, 46] and aligned original residues=[21, 43, 65] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).

• G001 — Use the complete integer congruence system with positive distinct moduli=[3, 5, 46] and aligned original residues=[21, 43, 65] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-chinese-remainder-theorem

Step-by-step derivation

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(21, 3)

original residue slot0 exact quotient = 7

original residue slot0 floor quotient multiple := mul(7, 3)

original residue slot0 floor quotient multiple = 21

original residue slot0 normalized remainder := sub(21, 21)

original residue slot0 normalized remainder = 0

original residue slot0:integer value=21;positive modulus=3;exact ratio=7;floor quotient=7

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

original residue slot1 exact quotient := div(43, 5)

original residue slot1 exact quotient = 43/5

original residue slot1 floor quotient multiple := mul(8, 5)

original residue slot1 floor quotient multiple = 40

original residue slot1 normalized remainder := sub(43, 40)

original residue slot1 normalized remainder = 3

original residue slot1:integer value=43;positive modulus=5;exact ratio=43/5;floor quotient=8

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

original residue slot2 exact quotient := div(65, 46)

original residue slot2 exact quotient = 65/46

original residue slot2 floor quotient multiple := mul(1, 46)

original residue slot2 floor quotient multiple = 46

original residue slot2 normalized remainder := sub(65, 46)

original residue slot2 normalized remainder = 19

original residue slot2:integer value=65;positive modulus=46;exact ratio=65/46;floor quotient=1

value=q*modulus+remainder;normalized residue=19 in[0,46)

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, 46)

CRT modulus product prefix2 = 690

LCM prefix2 gcd:positive original gcd inputs=[15, 46]

Euclidean rows[a,b,q,r] with a=q*b+r=[[15, 46, 0, 15], [46, 15, 3, 1], [15, 1, 15, 0]];positive gcd=1

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

CRT LCM prefix2 quotient = 15

CRT LCM prefix2 value := mul(15, 46)

CRT LCM prefix2 value = 690

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, 3)

CRT original pair0,1 residue difference = -3

CRT original pair0,1 difference divided by gcd := div(-3, 1)

CRT original pair0,1 difference divided by gcd = -3

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

difference/gcd=-3;pair coprime=Yes;pair compatible=Yes

original modulus pair0,2:positive original gcd inputs=[3, 46]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 46, 0, 3], [46, 3, 15, 1], [3, 1, 3, 0]];positive gcd=1

CRT original pair0,2 residue difference := sub(0, 19)

CRT original pair0,2 residue difference = -19

CRT original pair0,2 difference divided by gcd := div(-19, 1)

CRT original pair0,2 difference divided by gcd = -19

original constraint slots=[0, 2];moduli=[3, 46];gcd=1;normalized residue difference=-19

difference/gcd=-19;pair coprime=Yes;pair compatible=Yes

original modulus pair1,2:positive original gcd inputs=[5, 46]

Euclidean rows[a,b,q,r] with a=q*b+r=[[5, 46, 0, 5], [46, 5, 9, 1], [5, 1, 5, 0]];positive gcd=1

CRT original pair1,2 residue difference := sub(3, 19)

CRT original pair1,2 residue difference = -16

CRT original pair1,2 difference divided by gcd := div(-16, 1)

CRT original pair1,2 difference divided by gcd = -16

original constraint slots=[1, 2];moduli=[5, 46];gcd=1;normalized residue difference=-16

difference/gcd=-16;pair coprime=Yes;pair compatible=Yes

original aligned moduli=[3, 5, 46];original signed residues=[21, 43, 65];normalized residues=[0, 3, 19];all pair audits=[[0, 1, 1, Yes], [0, 2, 1, Yes], [1, 2, 1, Yes]]

pairwise coprime=Yes;gcd compatible=Yes;product=690;LCM period=690

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(3, 0)

merge slot1 new residue minus old class = 3

merge slot1 difference divided by gcd := div(3, 1)

merge slot1 difference divided by gcd = 3

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(3, 2)

merge slot1 parameter congruence product = 6

merge slot1 normalized parameter exact quotient := div(6, 5)

merge slot1 normalized parameter exact quotient = 6/5

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(6, 5)

merge slot1 normalized parameter normalized remainder = 1

merge slot1 normalized parameter:integer value=6;positive modulus=5;exact ratio=6/5;floor quotient=1

value=q*modulus+remainder;normalized residue=1 in[0,5)

merge slot1 old modulus times parameter := mul(3, 1)

merge slot1 old modulus times parameter = 3

merge slot1 unnormalized common solution := sum(0, 3)

merge slot1 unnormalized common solution = 3

merge slot1 least class representative exact quotient := div(3, 15)

merge slot1 least class representative exact quotient = 1/5

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(3, 0)

merge slot1 least class representative normalized remainder = 3

merge slot1 least class representative:integer value=3;positive modulus=15;exact ratio=1/5;floor quotient=0

value=q*modulus+remainder;normalized residue=3 in[0,15)

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

write old x=a+M*t;reduced equation gives t=1 mod5;ALL merged solutions x=3 mod15

merge original slot2 Euclidean step0 quotient fraction := div(15, 46)

merge original slot2 Euclidean step0 quotient fraction = 15/46

merge original slot2 Euclidean step0 divisor multiple := mul(0, 46)

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, 46];coefficient pairs=[[1, 0], [0, 1]];exact ratio=15/46;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(46, 15)

merge original slot2 Euclidean step1 quotient fraction = 46/15

merge original slot2 Euclidean step1 divisor multiple := mul(3, 15)

merge original slot2 Euclidean step1 divisor multiple = 45

merge original slot2 Euclidean step1 remainder := sub(46, 45)

merge original slot2 Euclidean step1 remainder = 1

merge original slot2 Euclidean step1 first coefficient multiple := mul(3, 1)

merge original slot2 Euclidean step1 first coefficient multiple = 3

merge original slot2 Euclidean step1 first coefficient difference := sub(0, 3)

merge original slot2 Euclidean step1 first coefficient difference = -3

merge original slot2 Euclidean step1 second coefficient multiple := mul(3, 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=[46, 15];coefficient pairs=[[0, 1], [1, 0]];exact ratio=46/15;floor q=3

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

merge original slot2 Euclidean step2 quotient fraction := div(15, 1)

merge original slot2 Euclidean step2 quotient fraction = 15

merge original slot2 Euclidean step2 divisor multiple := mul(15, 1)

merge original slot2 Euclidean step2 divisor multiple = 15

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

merge original slot2 Euclidean step2 remainder = 0

merge original slot2 Euclidean step2 first coefficient multiple := mul(15, -3)

merge original slot2 Euclidean step2 first coefficient multiple = -45

merge original slot2 Euclidean step2 first coefficient difference := sub(1, -45)

merge original slot2 Euclidean step2 first coefficient difference = 46

merge original slot2 Euclidean step2 second coefficient multiple := mul(15, 1)

merge original slot2 Euclidean step2 second coefficient multiple = 15

merge original slot2 Euclidean step2 second coefficient difference := sub(0, 15)

merge original slot2 Euclidean step2 second coefficient difference = -15

merge original slot2:step=2;current remainders=[15, 1];coefficient pairs=[[1, 0], [-3, 1]];exact ratio=15;floor q=15

next remainder=0;next signed coefficient pair=[46, -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(46, 1)

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(19, 3)

merge slot2 new residue minus old class = 16

merge slot2 difference divided by gcd := div(16, 1)

merge slot2 difference divided by gcd = 16

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

merge slot2 reduced old modulus = 15

merge slot2 reduced new modulus := div(46, 1)

merge slot2 reduced new modulus = 46

merge slot2 new LCM period := mul(15, 46)

merge slot2 new LCM period = 690

merge slot2 reduced-unit inverse exact quotient := div(-3, 46)

merge slot2 reduced-unit inverse exact quotient = -3/46

merge slot2 reduced-unit inverse floor quotient multiple := mul(-1, 46)

merge slot2 reduced-unit inverse floor quotient multiple = -46

merge slot2 reduced-unit inverse normalized remainder := sub(-3, -46)

merge slot2 reduced-unit inverse normalized remainder = 43

merge slot2 reduced-unit inverse:integer value=-3;positive modulus=46;exact ratio=-3/46;floor quotient=-1

value=q*modulus+remainder;normalized residue=43 in[0,46)

merge slot2 parameter congruence product := mul(16, 43)

merge slot2 parameter congruence product = 688

merge slot2 normalized parameter exact quotient := div(688, 46)

merge slot2 normalized parameter exact quotient = 344/23

merge slot2 normalized parameter floor quotient multiple := mul(14, 46)

merge slot2 normalized parameter floor quotient multiple = 644

merge slot2 normalized parameter normalized remainder := sub(688, 644)

merge slot2 normalized parameter normalized remainder = 44

merge slot2 normalized parameter:integer value=688;positive modulus=46;exact ratio=344/23;floor quotient=14

value=q*modulus+remainder;normalized residue=44 in[0,46)

merge slot2 old modulus times parameter := mul(15, 44)

merge slot2 old modulus times parameter = 660

merge slot2 unnormalized common solution := sum(3, 660)

merge slot2 unnormalized common solution = 663

merge slot2 least class representative exact quotient := div(663, 690)

merge slot2 least class representative exact quotient = 221/230

merge slot2 least class representative floor quotient multiple := mul(0, 690)

merge slot2 least class representative floor quotient multiple = 0

merge slot2 least class representative normalized remainder := sub(663, 0)

merge slot2 least class representative normalized remainder = 663

merge slot2 least class representative:integer value=663;positive modulus=690;exact ratio=221/230;floor quotient=0

value=q*modulus+remainder;normalized residue=663 in[0,690)

old FULL solution class=3 mod15;new original constraint=19 mod46;gcd=1;Bezout coefficients=[-3, 1];reduced coprime moduli=[15, 46]

write old x=a+M*t;reduced equation gives t=44 mod46;ALL merged solutions x=663 mod690

Reverse checks

• S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S180 · premise_and_result_audit — Recompute merge slot2 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute merge slot2 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute merge slot2 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute merge slot2 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute merge slot2 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S174 · premise_and_result_audit — Recompute merge slot2 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute merge slot2 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute merge slot2 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute merge slot2 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute merge slot2 new LCM period with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Recompute merge slot2 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute merge slot2 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute merge slot2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute merge slot2 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute merge original slot2 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute merge original slot2 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute merge original slot2 final first Bezout term with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 quotient fraction with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute merge slot1 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute merge slot1 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute merge slot1 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute merge slot1 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute merge slot1 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Recompute merge slot1 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute merge slot1 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute merge slot1 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute merge slot1 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute merge slot1 new LCM period with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute merge slot1 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute merge slot1 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute merge slot1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute merge slot1 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute merge original slot1 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute merge original slot1 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute merge original slot1 final first Bezout term with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 remainder with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 divisor multiple with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 quotient fraction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 quotient fraction with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Recompute merge slot0 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute merge slot0 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute merge slot0 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute merge slot0 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute merge slot0 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute merge slot0 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute merge slot0 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute merge slot0 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute merge slot0 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute merge slot0 new LCM period with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute merge slot0 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute merge slot0 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute merge slot0 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute merge slot0 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute merge original slot0 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute merge original slot0 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute merge original slot0 final first Bezout term with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute CRT original pair1,2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute CRT original pair1,2 residue difference with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute CRT original pair0,2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute CRT original pair0,2 residue difference with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute CRT original pair0,1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute CRT original pair0,1 residue difference with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute CRT LCM prefix2 value with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute CRT LCM prefix2 quotient with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute CRT modulus product prefix2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute CRT LCM prefix1 value with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute CRT LCM prefix1 quotient with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute CRT modulus product prefix1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute CRT LCM prefix0 value with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute CRT LCM prefix0 quotient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute CRT modulus product prefix0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute original residue slot2 normalized remainder with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute original residue slot2 floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute original residue slot2 exact quotient with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute original residue slot1 normalized remainder with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute original residue slot1 floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute original residue slot1 exact quotient with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute original residue slot0 normalized remainder with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute original residue slot0 floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute original residue slot0 exact quotient with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([3, 5, 46], [0, 3, 19], 663, 690, [Yes, Yes, Yes], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.