Question 18074: 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, 49] and aligned original residues=[23, 47, 71] 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, 49] and aligned original residues=[23, 47, 71] 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(23, 3)

original residue slot0 exact quotient = 23/3

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

original residue slot0 floor quotient multiple = 21

original residue slot0 normalized remainder := sub(23, 21)

original residue slot0 normalized remainder = 2

original residue slot0:integer value=23;positive modulus=3;exact ratio=23/3;floor quotient=7

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

original residue slot1 exact quotient := div(47, 5)

original residue slot1 exact quotient = 47/5

original residue slot1 floor quotient multiple := mul(9, 5)

original residue slot1 floor quotient multiple = 45

original residue slot1 normalized remainder := sub(47, 45)

original residue slot1 normalized remainder = 2

original residue slot1:integer value=47;positive modulus=5;exact ratio=47/5;floor quotient=9

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

original residue slot2 exact quotient := div(71, 49)

original residue slot2 exact quotient = 71/49

original residue slot2 floor quotient multiple := mul(1, 49)

original residue slot2 floor quotient multiple = 49

original residue slot2 normalized remainder := sub(71, 49)

original residue slot2 normalized remainder = 22

original residue slot2:integer value=71;positive modulus=49;exact ratio=71/49;floor quotient=1

value=q*modulus+remainder;normalized residue=22 in[0,49)

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, 49)

CRT modulus product prefix2 = 735

LCM prefix2 gcd:positive original gcd inputs=[15, 49]

Euclidean rows[a,b,q,r] with a=q*b+r=[[15, 49, 0, 15], [49, 15, 3, 4], [15, 4, 3, 3], [4, 3, 1, 1], [3, 1, 3, 0]];positive gcd=1

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

CRT LCM prefix2 quotient = 15

CRT LCM prefix2 value := mul(15, 49)

CRT LCM prefix2 value = 735

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(2, 2)

CRT original pair0,1 residue difference = 0

CRT original pair0,1 difference divided by gcd := div(0, 1)

CRT original pair0,1 difference divided by gcd = 0

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

difference/gcd=0;pair coprime=Yes;pair compatible=Yes

original modulus pair0,2:positive original gcd inputs=[3, 49]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 49, 0, 3], [49, 3, 16, 1], [3, 1, 3, 0]];positive gcd=1

CRT original pair0,2 residue difference := sub(2, 22)

CRT original pair0,2 residue difference = -20

CRT original pair0,2 difference divided by gcd := div(-20, 1)

CRT original pair0,2 difference divided by gcd = -20

original constraint slots=[0, 2];moduli=[3, 49];gcd=1;normalized residue difference=-20

difference/gcd=-20;pair coprime=Yes;pair compatible=Yes

original modulus pair1,2:positive original gcd inputs=[5, 49]

Euclidean rows[a,b,q,r] with a=q*b+r=[[5, 49, 0, 5], [49, 5, 9, 4], [5, 4, 1, 1], [4, 1, 4, 0]];positive gcd=1

CRT original pair1,2 residue difference := sub(2, 22)

CRT original pair1,2 residue difference = -20

CRT original pair1,2 difference divided by gcd := div(-20, 1)

CRT original pair1,2 difference divided by gcd = -20

original constraint slots=[1, 2];moduli=[5, 49];gcd=1;normalized residue difference=-20

difference/gcd=-20;pair coprime=Yes;pair compatible=Yes

original aligned moduli=[3, 5, 49];original signed residues=[23, 47, 71];normalized residues=[2, 2, 22];all pair audits=[[0, 1, 1, Yes], [0, 2, 1, Yes], [1, 2, 1, Yes]]

pairwise coprime=Yes;gcd compatible=Yes;product=735;LCM period=735

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(2, 0)

merge slot0 new residue minus old class = 2

merge slot0 difference divided by gcd := div(2, 1)

merge slot0 difference divided by gcd = 2

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(2, 1)

merge slot0 parameter congruence product = 2

merge slot0 normalized parameter exact quotient := div(2, 3)

merge slot0 normalized parameter exact quotient = 2/3

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

merge slot0 normalized parameter floor quotient multiple = 0

merge slot0 normalized parameter normalized remainder := sub(2, 0)

merge slot0 normalized parameter normalized remainder = 2

merge slot0 normalized parameter:integer value=2;positive modulus=3;exact ratio=2/3;floor quotient=0

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

merge slot0 old modulus times parameter := mul(1, 2)

merge slot0 old modulus times parameter = 2

merge slot0 unnormalized common solution := sum(0, 2)

merge slot0 unnormalized common solution = 2

merge slot0 least class representative exact quotient := div(2, 3)

merge slot0 least class representative exact quotient = 2/3

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

merge slot0 least class representative floor quotient multiple = 0

merge slot0 least class representative normalized remainder := sub(2, 0)

merge slot0 least class representative normalized remainder = 2

merge slot0 least class representative:integer value=2;positive modulus=3;exact ratio=2/3;floor quotient=0

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

old FULL solution class=0 mod1;new original constraint=2 mod3;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 3]

write old x=a+M*t;reduced equation gives t=2 mod3;ALL merged solutions x=2 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(2, 2)

merge slot1 new residue minus old class = 0

merge slot1 difference divided by gcd := div(0, 1)

merge slot1 difference divided by gcd = 0

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(0, 2)

merge slot1 parameter congruence product = 0

merge slot1 normalized parameter exact quotient := div(0, 5)

merge slot1 normalized parameter exact quotient = 0

merge slot1 normalized parameter floor quotient multiple := mul(0, 5)

merge slot1 normalized parameter floor quotient multiple = 0

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

merge slot1 normalized parameter normalized remainder = 0

merge slot1 normalized parameter:integer value=0;positive modulus=5;exact ratio=0;floor quotient=0

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

merge slot1 old modulus times parameter := mul(3, 0)

merge slot1 old modulus times parameter = 0

merge slot1 unnormalized common solution := sum(2, 0)

merge slot1 unnormalized common solution = 2

merge slot1 least class representative exact quotient := div(2, 15)

merge slot1 least class representative exact quotient = 2/15

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(2, 0)

merge slot1 least class representative normalized remainder = 2

merge slot1 least class representative:integer value=2;positive modulus=15;exact ratio=2/15;floor quotient=0

value=q*modulus+remainder;normalized residue=2 in[0,15)

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

write old x=a+M*t;reduced equation gives t=0 mod5;ALL merged solutions x=2 mod15

merge original slot2 Euclidean step0 quotient fraction := div(15, 49)

merge original slot2 Euclidean step0 quotient fraction = 15/49

merge original slot2 Euclidean step0 divisor multiple := mul(0, 49)

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, 49];coefficient pairs=[[1, 0], [0, 1]];exact ratio=15/49;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(49, 15)

merge original slot2 Euclidean step1 quotient fraction = 49/15

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(49, 45)

merge original slot2 Euclidean step1 remainder = 4

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=[49, 15];coefficient pairs=[[0, 1], [1, 0]];exact ratio=49/15;floor q=3

next remainder=4;next signed coefficient pair=[-3, 1];a*x+b*y equals next remainder

merge original slot2 Euclidean step2 quotient fraction := div(15, 4)

merge original slot2 Euclidean step2 quotient fraction = 15/4

merge original slot2 Euclidean step2 divisor multiple := mul(3, 4)

merge original slot2 Euclidean step2 divisor multiple = 12

merge original slot2 Euclidean step2 remainder := sub(15, 12)

merge original slot2 Euclidean step2 remainder = 3

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

merge original slot2 Euclidean step2 first coefficient multiple = -9

merge original slot2 Euclidean step2 first coefficient difference := sub(1, -9)

merge original slot2 Euclidean step2 first coefficient difference = 10

merge original slot2 Euclidean step2 second coefficient multiple := mul(3, 1)

merge original slot2 Euclidean step2 second coefficient multiple = 3

merge original slot2 Euclidean step2 second coefficient difference := sub(0, 3)

merge original slot2 Euclidean step2 second coefficient difference = -3

merge original slot2:step=2;current remainders=[15, 4];coefficient pairs=[[1, 0], [-3, 1]];exact ratio=15/4;floor q=3

next remainder=3;next signed coefficient pair=[10, -3];a*x+b*y equals next remainder

merge original slot2 Euclidean step3 quotient fraction := div(4, 3)

merge original slot2 Euclidean step3 quotient fraction = 4/3

merge original slot2 Euclidean step3 divisor multiple := mul(1, 3)

merge original slot2 Euclidean step3 divisor multiple = 3

merge original slot2 Euclidean step3 remainder := sub(4, 3)

merge original slot2 Euclidean step3 remainder = 1

merge original slot2 Euclidean step3 first coefficient multiple := mul(1, 10)

merge original slot2 Euclidean step3 first coefficient multiple = 10

merge original slot2 Euclidean step3 first coefficient difference := sub(-3, 10)

merge original slot2 Euclidean step3 first coefficient difference = -13

merge original slot2 Euclidean step3 second coefficient multiple := mul(1, -3)

merge original slot2 Euclidean step3 second coefficient multiple = -3

merge original slot2 Euclidean step3 second coefficient difference := sub(1, -3)

merge original slot2 Euclidean step3 second coefficient difference = 4

merge original slot2:step=3;current remainders=[4, 3];coefficient pairs=[[-3, 1], [10, -3]];exact ratio=4/3;floor q=1

next remainder=1;next signed coefficient pair=[-13, 4];a*x+b*y equals next remainder

merge original slot2 Euclidean step4 quotient fraction := div(3, 1)

merge original slot2 Euclidean step4 quotient fraction = 3

merge original slot2 Euclidean step4 divisor multiple := mul(3, 1)

merge original slot2 Euclidean step4 divisor multiple = 3

merge original slot2 Euclidean step4 remainder := sub(3, 3)

merge original slot2 Euclidean step4 remainder = 0

merge original slot2 Euclidean step4 first coefficient multiple := mul(3, -13)

merge original slot2 Euclidean step4 first coefficient multiple = -39

merge original slot2 Euclidean step4 first coefficient difference := sub(10, -39)

merge original slot2 Euclidean step4 first coefficient difference = 49

merge original slot2 Euclidean step4 second coefficient multiple := mul(3, 4)

merge original slot2 Euclidean step4 second coefficient multiple = 12

merge original slot2 Euclidean step4 second coefficient difference := sub(-3, 12)

merge original slot2 Euclidean step4 second coefficient difference = -15

merge original slot2:step=4;current remainders=[3, 1];coefficient pairs=[[10, -3], [-13, 4]];exact ratio=3;floor q=3

next remainder=0;next signed coefficient pair=[49, -15];a*x+b*y equals next remainder

merge original slot2 final first Bezout term := mul(15, -13)

merge original slot2 final first Bezout term = -195

merge original slot2 final second Bezout term := mul(49, 4)

merge original slot2 final second Bezout term = 196

merge original slot2 final Bezout identity := sum(-195, 196)

merge original slot2 final Bezout identity = 1

merge slot2 new residue minus old class := sub(22, 2)

merge slot2 new residue minus old class = 20

merge slot2 difference divided by gcd := div(20, 1)

merge slot2 difference divided by gcd = 20

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

merge slot2 reduced old modulus = 15

merge slot2 reduced new modulus := div(49, 1)

merge slot2 reduced new modulus = 49

merge slot2 new LCM period := mul(15, 49)

merge slot2 new LCM period = 735

merge slot2 reduced-unit inverse exact quotient := div(-13, 49)

merge slot2 reduced-unit inverse exact quotient = -13/49

merge slot2 reduced-unit inverse floor quotient multiple := mul(-1, 49)

merge slot2 reduced-unit inverse floor quotient multiple = -49

merge slot2 reduced-unit inverse normalized remainder := sub(-13, -49)

merge slot2 reduced-unit inverse normalized remainder = 36

merge slot2 reduced-unit inverse:integer value=-13;positive modulus=49;exact ratio=-13/49;floor quotient=-1

value=q*modulus+remainder;normalized residue=36 in[0,49)

merge slot2 parameter congruence product := mul(20, 36)

merge slot2 parameter congruence product = 720

merge slot2 normalized parameter exact quotient := div(720, 49)

merge slot2 normalized parameter exact quotient = 720/49

merge slot2 normalized parameter floor quotient multiple := mul(14, 49)

merge slot2 normalized parameter floor quotient multiple = 686

merge slot2 normalized parameter normalized remainder := sub(720, 686)

merge slot2 normalized parameter normalized remainder = 34

merge slot2 normalized parameter:integer value=720;positive modulus=49;exact ratio=720/49;floor quotient=14

value=q*modulus+remainder;normalized residue=34 in[0,49)

merge slot2 old modulus times parameter := mul(15, 34)

merge slot2 old modulus times parameter = 510

merge slot2 unnormalized common solution := sum(2, 510)

merge slot2 unnormalized common solution = 512

merge slot2 least class representative exact quotient := div(512, 735)

merge slot2 least class representative exact quotient = 512/735

merge slot2 least class representative floor quotient multiple := mul(0, 735)

merge slot2 least class representative floor quotient multiple = 0

merge slot2 least class representative normalized remainder := sub(512, 0)

merge slot2 least class representative normalized remainder = 512

merge slot2 least class representative:integer value=512;positive modulus=735;exact ratio=512/735;floor quotient=0

value=q*modulus+remainder;normalized residue=512 in[0,735)

old FULL solution class=2 mod15;new original constraint=22 mod49;gcd=1;Bezout coefficients=[-13, 4];reduced coprime moduli=[15, 49]

write old x=a+M*t;reduced equation gives t=34 mod49;ALL merged solutions x=512 mod735

Reverse checks

• S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S196 · premise_and_result_audit — Recompute merge slot2 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Recompute merge slot2 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S194 · premise_and_result_audit — Recompute merge slot2 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute merge slot2 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Recompute merge slot2 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S190 · premise_and_result_audit — Recompute merge slot2 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Recompute merge slot2 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S188 · premise_and_result_audit — Recompute merge slot2 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute merge slot2 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute merge slot2 new LCM period with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute merge slot2 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute merge slot2 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute merge slot2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute merge slot2 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute merge original slot2 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute merge original slot2 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute merge original slot2 final first Bezout term with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S173 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 remainder with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 divisor multiple with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 quotient fraction with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 remainder with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 divisor multiple with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 quotient fraction 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, 49], [2, 2, 22], 512, 735, [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.