Question 18035: Solve a Pairwise-Coprime Pair after Normalizing Both Residues

Original question and givens

Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] 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. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).

• G001 — Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] 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. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, 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-pair;original snapshot and mathematics retained

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

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

original residue slot0 exact quotient = 3

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

original residue slot0 floor quotient multiple = 9

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

original residue slot0 normalized remainder = 0

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

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

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

original residue slot1 exact quotient = 1

original residue slot1 floor quotient multiple := mul(1, 19)

original residue slot1 floor quotient multiple = 19

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

original residue slot1 normalized remainder = 0

original residue slot1:integer value=19;positive modulus=19;exact ratio=1;floor quotient=1

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

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, 19)

CRT modulus product prefix1 = 57

LCM prefix1 gcd:positive original gcd inputs=[3, 19]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 19, 0, 3], [19, 3, 6, 1], [3, 1, 3, 0]];positive gcd=1

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

CRT LCM prefix1 quotient = 3

CRT LCM prefix1 value := mul(3, 19)

CRT LCM prefix1 value = 57

original modulus pair0,1:positive original gcd inputs=[3, 19]

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 19, 0, 3], [19, 3, 6, 1], [3, 1, 3, 0]];positive gcd=1

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

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, 19];gcd=1;normalized residue difference=0

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

original aligned moduli=[3, 19];original signed residues=[9, 19];normalized residues=[0, 0];all pair audits=[[0, 1, 1, Yes]]

pairwise coprime=Yes;gcd compatible=Yes;product=57;LCM period=57

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, 19)

merge original slot1 Euclidean step0 quotient fraction = 3/19

merge original slot1 Euclidean step0 divisor multiple := mul(0, 19)

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, 19];coefficient pairs=[[1, 0], [0, 1]];exact ratio=3/19;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(19, 3)

merge original slot1 Euclidean step1 quotient fraction = 19/3

merge original slot1 Euclidean step1 divisor multiple := mul(6, 3)

merge original slot1 Euclidean step1 divisor multiple = 18

merge original slot1 Euclidean step1 remainder := sub(19, 18)

merge original slot1 Euclidean step1 remainder = 1

merge original slot1 Euclidean step1 first coefficient multiple := mul(6, 1)

merge original slot1 Euclidean step1 first coefficient multiple = 6

merge original slot1 Euclidean step1 first coefficient difference := sub(0, 6)

merge original slot1 Euclidean step1 first coefficient difference = -6

merge original slot1 Euclidean step1 second coefficient multiple := mul(6, 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=[19, 3];coefficient pairs=[[0, 1], [1, 0]];exact ratio=19/3;floor q=6

next remainder=1;next signed coefficient pair=[-6, 1];a*x+b*y equals next remainder

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

merge original slot1 Euclidean step2 quotient fraction = 3

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

merge original slot1 Euclidean step2 divisor multiple = 3

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

merge original slot1 Euclidean step2 remainder = 0

merge original slot1 Euclidean step2 first coefficient multiple := mul(3, -6)

merge original slot1 Euclidean step2 first coefficient multiple = -18

merge original slot1 Euclidean step2 first coefficient difference := sub(1, -18)

merge original slot1 Euclidean step2 first coefficient difference = 19

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

merge original slot1 Euclidean step2 second coefficient multiple = 3

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

merge original slot1 Euclidean step2 second coefficient difference = -3

merge original slot1:step=2;current remainders=[3, 1];coefficient pairs=[[1, 0], [-6, 1]];exact ratio=3;floor q=3

next remainder=0;next signed coefficient pair=[19, -3];a*x+b*y equals next remainder

merge original slot1 final first Bezout term := mul(3, -6)

merge original slot1 final first Bezout term = -18

merge original slot1 final second Bezout term := mul(19, 1)

merge original slot1 final second Bezout term = 19

merge original slot1 final Bezout identity := sum(-18, 19)

merge original slot1 final Bezout identity = 1

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

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(19, 1)

merge slot1 reduced new modulus = 19

merge slot1 new LCM period := mul(3, 19)

merge slot1 new LCM period = 57

merge slot1 reduced-unit inverse exact quotient := div(-6, 19)

merge slot1 reduced-unit inverse exact quotient = -6/19

merge slot1 reduced-unit inverse floor quotient multiple := mul(-1, 19)

merge slot1 reduced-unit inverse floor quotient multiple = -19

merge slot1 reduced-unit inverse normalized remainder := sub(-6, -19)

merge slot1 reduced-unit inverse normalized remainder = 13

merge slot1 reduced-unit inverse:integer value=-6;positive modulus=19;exact ratio=-6/19;floor quotient=-1

value=q*modulus+remainder;normalized residue=13 in[0,19)

merge slot1 parameter congruence product := mul(0, 13)

merge slot1 parameter congruence product = 0

merge slot1 normalized parameter exact quotient := div(0, 19)

merge slot1 normalized parameter exact quotient = 0

merge slot1 normalized parameter floor quotient multiple := mul(0, 19)

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=19;exact ratio=0;floor quotient=0

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

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

merge slot1 old modulus times parameter = 0

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

merge slot1 unnormalized common solution = 0

merge slot1 least class representative exact quotient := div(0, 57)

merge slot1 least class representative exact quotient = 0

merge slot1 least class representative floor quotient multiple := mul(0, 57)

merge slot1 least class representative floor quotient multiple = 0

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

merge slot1 least class representative normalized remainder = 0

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

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

old FULL solution class=0 mod3;new original constraint=0 mod19;gcd=1;Bezout coefficients=[-6, 1];reduced coprime moduli=[3, 19]

write old x=a+M*t;reduced equation gives t=0 mod19;ALL merged solutions x=0 mod57

Reverse checks

• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Recompute merge slot1 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute merge slot1 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute merge slot1 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute merge slot1 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute merge slot1 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Recompute merge slot1 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute merge slot1 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute merge slot1 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute merge slot1 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute merge slot1 new LCM period with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute merge slot1 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute merge slot1 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute merge slot1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute merge slot1 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute merge original slot1 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute merge original slot1 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute merge original slot1 final first Bezout term 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 step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 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 — Recompute merge original slot1 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute merge slot0 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute merge slot0 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute merge slot0 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute merge slot0 unnormalized common solution with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute merge slot0 old modulus times parameter with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Recompute merge slot0 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute merge slot0 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute merge slot0 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute merge slot0 parameter congruence product with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute merge slot0 new LCM period with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute merge slot0 reduced new modulus with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute merge slot0 reduced old modulus with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute merge slot0 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute merge slot0 new residue minus old class with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute merge original slot0 final Bezout identity with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute merge original slot0 final second Bezout term with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute merge original slot0 final first Bezout term 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 — Recompute merge original slot0 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 quotient fraction 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 — Recompute merge original slot0 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute CRT original pair0,1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute CRT original pair0,1 residue difference with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute CRT LCM prefix1 value with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute CRT LCM prefix1 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 prefix1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute CRT LCM prefix0 value with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute CRT LCM prefix0 quotient with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute CRT modulus product prefix0 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, 19], [0, 0], 0, 57, 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.