# Question 18047: Solve a Pairwise-Coprime Pair after Normalizing Both Residues [Return to catalogue page 91](https://ocme.evemisslab.com/en/questions/page/91/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations/#q-20261014-advance-20261104-r05-chinese-remainder-theorem-0022) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0035-18047 · q-20261014-advance-20261104-r05-chinese-remainder-theorem-0022 ## Original question and givens Use the complete integer congruence system with positive distinct moduli=[3, 37] and aligned original residues=[21, 43] 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, 37] and aligned original residues=[21, 43] 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](https://ocme.evemisslab.com/en/mko/mko-chinese-remainder-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=crt-solve-pair;original snapshot and mathematics retained After explicit ordered source output fields=[moduli, normalized_residues, solution, product, solved] Conditions: none Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore original residue slot0 exact quotient := div(21, 3) After original residue slot0 exact quotient = 7 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore original residue slot0 floor quotient multiple := mul(7, 3) After original residue slot0 floor quotient multiple = 21 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original residue slot0 normalized remainder := sub(21, 21) After original residue slot0 normalized remainder = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore original residue slot0:integer value=21;positive modulus=3;exact ratio=7;floor quotient=7 After value=q*modulus+remainder;normalized residue=0 in[0,3) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S004 S006 · calculationBefore original residue slot1 exact quotient := div(43, 37) After original residue slot1 exact quotient = 43/37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore original residue slot1 floor quotient multiple := mul(1, 37) After original residue slot1 floor quotient multiple = 37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore original residue slot1 normalized remainder := sub(43, 37) After original residue slot1 normalized remainder = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore original residue slot1:integer value=43;positive modulus=37;exact ratio=43/37;floor quotient=1 After value=q*modulus+remainder;normalized residue=6 in[0,37) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S008 S010 · calculationBefore CRT modulus product prefix0 := mul(1, 3) After CRT modulus product prefix0 = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore LCM prefix0 gcd:positive original gcd inputs=[1, 3] After Euclidean rows[a,b,q,r] with a=q*b+r=[[1, 3, 0, 1], [3, 1, 3, 0]];positive gcd=1 Conditions: none Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking. Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S010 S012 · calculationBefore CRT LCM prefix0 quotient := div(1, 1) After CRT LCM prefix0 quotient = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore CRT LCM prefix0 value := mul(1, 3) After CRT LCM prefix0 value = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore CRT modulus product prefix1 := mul(3, 37) After CRT modulus product prefix1 = 111 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore LCM prefix1 gcd:positive original gcd inputs=[3, 37] After Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 37, 0, 3], [37, 3, 12, 1], [3, 1, 3, 0]];positive gcd=1 Conditions: none Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking. Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S014 S016 · calculationBefore CRT LCM prefix1 quotient := div(3, 1) After CRT LCM prefix1 quotient = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore CRT LCM prefix1 value := mul(3, 37) After CRT LCM prefix1 value = 111 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore original modulus pair0,1:positive original gcd inputs=[3, 37] After Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 37, 0, 3], [37, 3, 12, 1], [3, 1, 3, 0]];positive gcd=1 Conditions: none Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking. Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S017 S019 · calculationBefore CRT original pair0,1 residue difference := sub(0, 6) After CRT original pair0,1 residue difference = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore CRT original pair0,1 difference divided by gcd := div(-6, 1) After CRT original pair0,1 difference divided by gcd = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore original constraint slots=[0, 1];moduli=[3, 37];gcd=1;normalized residue difference=-6 After difference/gcd=-6;pair coprime=Yes;pair compatible=Yes Conditions: none A common solution forces each residue difference to be divisible by the corresponding gcd.Check every pair;noncoprimality is not automatic inconsistency. Step identifiers and dependenciesRule: crt_every_pair_compatibility_not_automatic_inconsistency · Uses: G001, S020 S022 · impliesBefore original aligned moduli=[3, 37];original signed residues=[21, 43];normalized residues=[0, 6];all pair audits=[[0, 1, 1, Yes]] After pairwise coprime=Yes;gcd compatible=Yes;product=111;LCM period=111 Conditions: none Retain original aligned constraint order,not separate semantic sorting.Main coprime operations use the product;the noncoprime boundary retains the lcm.Modulus1 adds no restriction. Step identifiers and dependenciesRule: crt_original_constraint_order_product_vs_lcm · Uses: G001, S021 S023 · calculationBefore merge original slot0 Euclidean step0 quotient fraction := div(1, 3) After merge original slot0 Euclidean step0 quotient fraction = 1/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore merge original slot0 Euclidean step0 divisor multiple := mul(0, 3) After merge original slot0 Euclidean step0 divisor multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore merge original slot0 Euclidean step0 remainder := sub(1, 0) After merge original slot0 Euclidean step0 remainder = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore merge original slot0 Euclidean step0 first coefficient multiple := mul(0, 0) After merge original slot0 Euclidean step0 first coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore merge original slot0 Euclidean step0 first coefficient difference := sub(1, 0) After merge original slot0 Euclidean step0 first coefficient difference = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore merge original slot0 Euclidean step0 second coefficient multiple := mul(0, 1) After merge original slot0 Euclidean step0 second coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore merge original slot0 Euclidean step0 second coefficient difference := sub(0, 0) After merge original slot0 Euclidean step0 second coefficient difference = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore merge original slot0:step=0;current remainders=[1, 3];coefficient pairs=[[1, 0], [0, 1]];exact ratio=1/3;floor q=0 After next remainder=1;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder Conditions: none Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse. Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S029 S031 · calculationBefore merge original slot0 Euclidean step1 quotient fraction := div(3, 1) After merge original slot0 Euclidean step1 quotient fraction = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore merge original slot0 Euclidean step1 divisor multiple := mul(3, 1) After merge original slot0 Euclidean step1 divisor multiple = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore merge original slot0 Euclidean step1 remainder := sub(3, 3) After merge original slot0 Euclidean step1 remainder = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore merge original slot0 Euclidean step1 first coefficient multiple := mul(3, 1) After merge original slot0 Euclidean step1 first coefficient multiple = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore merge original slot0 Euclidean step1 first coefficient difference := sub(0, 3) After merge original slot0 Euclidean step1 first coefficient difference = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore merge original slot0 Euclidean step1 second coefficient multiple := mul(3, 0) After merge original slot0 Euclidean step1 second coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore merge original slot0 Euclidean step1 second coefficient difference := sub(1, 0) After merge original slot0 Euclidean step1 second coefficient difference = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore merge original slot0:step=1;current remainders=[3, 1];coefficient pairs=[[0, 1], [1, 0]];exact ratio=3;floor q=3 After next remainder=0;next signed coefficient pair=[-3, 1];a*x+b*y equals next remainder Conditions: none Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse. Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S037 S039 · calculationBefore merge original slot0 final first Bezout term := mul(1, 1) After merge original slot0 final first Bezout term = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore merge original slot0 final second Bezout term := mul(3, 0) After merge original slot0 final second Bezout term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore merge original slot0 final Bezout identity := sum(1, 0) After merge original slot0 final Bezout identity = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore merge slot0 new residue minus old class := sub(0, 0) After merge slot0 new residue minus old class = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore merge slot0 difference divided by gcd := div(0, 1) After merge slot0 difference divided by gcd = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore merge slot0 reduced old modulus := div(1, 1) After merge slot0 reduced old modulus = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore merge slot0 reduced new modulus := div(3, 1) After merge slot0 reduced new modulus = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore merge slot0 new LCM period := mul(1, 3) After merge slot0 new LCM period = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore merge slot0 reduced-unit inverse exact quotient := div(1, 3) After merge slot0 reduced-unit inverse exact quotient = 1/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore merge slot0 reduced-unit inverse floor quotient multiple := mul(0, 3) After merge slot0 reduced-unit inverse floor quotient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore merge slot0 reduced-unit inverse normalized remainder := sub(1, 0) After merge slot0 reduced-unit inverse normalized remainder = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · impliesBefore merge slot0 reduced-unit inverse:integer value=1;positive modulus=3;exact ratio=1/3;floor quotient=0 After value=q*modulus+remainder;normalized residue=1 in[0,3) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S049 S051 · calculationBefore merge slot0 parameter congruence product := mul(0, 1) After merge slot0 parameter congruence product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore merge slot0 normalized parameter exact quotient := div(0, 3) After merge slot0 normalized parameter exact quotient = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore merge slot0 normalized parameter floor quotient multiple := mul(0, 3) After merge slot0 normalized parameter floor quotient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore merge slot0 normalized parameter normalized remainder := sub(0, 0) After merge slot0 normalized parameter normalized remainder = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · impliesBefore merge slot0 normalized parameter:integer value=0;positive modulus=3;exact ratio=0;floor quotient=0 After value=q*modulus+remainder;normalized residue=0 in[0,3) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S054 S056 · calculationBefore merge slot0 old modulus times parameter := mul(1, 0) After merge slot0 old modulus times parameter = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore merge slot0 unnormalized common solution := sum(0, 0) After merge slot0 unnormalized common solution = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore merge slot0 least class representative exact quotient := div(0, 3) After merge slot0 least class representative exact quotient = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore merge slot0 least class representative floor quotient multiple := mul(0, 3) After merge slot0 least class representative floor quotient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore merge slot0 least class representative normalized remainder := sub(0, 0) After merge slot0 least class representative normalized remainder = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore merge slot0 least class representative:integer value=0;positive modulus=3;exact ratio=0;floor quotient=0 After value=q*modulus+remainder;normalized residue=0 in[0,3) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S060 S062 · impliesBefore old FULL solution class=0 mod1;new original constraint=0 mod3;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 3] After write old x=a+M*t;reduced equation gives t=0 mod3;ALL merged solutions x=0 mod3 Conditions: none Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point. Step identifiers and dependenciesRule: crt_constructive_merge_all_solutions_not_one_witness · Uses: G001, S061 S063 · calculationBefore merge original slot1 Euclidean step0 quotient fraction := div(3, 37) After merge original slot1 Euclidean step0 quotient fraction = 3/37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore merge original slot1 Euclidean step0 divisor multiple := mul(0, 37) After merge original slot1 Euclidean step0 divisor multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore merge original slot1 Euclidean step0 remainder := sub(3, 0) After merge original slot1 Euclidean step0 remainder = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore merge original slot1 Euclidean step0 first coefficient multiple := mul(0, 0) After merge original slot1 Euclidean step0 first coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore merge original slot1 Euclidean step0 first coefficient difference := sub(1, 0) After merge original slot1 Euclidean step0 first coefficient difference = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore merge original slot1 Euclidean step0 second coefficient multiple := mul(0, 1) After merge original slot1 Euclidean step0 second coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore merge original slot1 Euclidean step0 second coefficient difference := sub(0, 0) After merge original slot1 Euclidean step0 second coefficient difference = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · impliesBefore merge original slot1:step=0;current remainders=[3, 37];coefficient pairs=[[1, 0], [0, 1]];exact ratio=3/37;floor q=0 After next remainder=3;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder Conditions: none Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse. Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S069 S071 · calculationBefore merge original slot1 Euclidean step1 quotient fraction := div(37, 3) After merge original slot1 Euclidean step1 quotient fraction = 37/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore merge original slot1 Euclidean step1 divisor multiple := mul(12, 3) After merge original slot1 Euclidean step1 divisor multiple = 36 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore merge original slot1 Euclidean step1 remainder := sub(37, 36) After merge original slot1 Euclidean step1 remainder = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore merge original slot1 Euclidean step1 first coefficient multiple := mul(12, 1) After merge original slot1 Euclidean step1 first coefficient multiple = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore merge original slot1 Euclidean step1 first coefficient difference := sub(0, 12) After merge original slot1 Euclidean step1 first coefficient difference = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore merge original slot1 Euclidean step1 second coefficient multiple := mul(12, 0) After merge original slot1 Euclidean step1 second coefficient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore merge original slot1 Euclidean step1 second coefficient difference := sub(1, 0) After merge original slot1 Euclidean step1 second coefficient difference = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore merge original slot1:step=1;current remainders=[37, 3];coefficient pairs=[[0, 1], [1, 0]];exact ratio=37/3;floor q=12 After next remainder=1;next signed coefficient pair=[-12, 1];a*x+b*y equals next remainder Conditions: none Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse. Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S077 S079 · calculationBefore merge original slot1 Euclidean step2 quotient fraction := div(3, 1) After merge original slot1 Euclidean step2 quotient fraction = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore merge original slot1 Euclidean step2 divisor multiple := mul(3, 1) After merge original slot1 Euclidean step2 divisor multiple = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore merge original slot1 Euclidean step2 remainder := sub(3, 3) After merge original slot1 Euclidean step2 remainder = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore merge original slot1 Euclidean step2 first coefficient multiple := mul(3, -12) After merge original slot1 Euclidean step2 first coefficient multiple = -36 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore merge original slot1 Euclidean step2 first coefficient difference := sub(1, -36) After merge original slot1 Euclidean step2 first coefficient difference = 37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore merge original slot1 Euclidean step2 second coefficient multiple := mul(3, 1) After merge original slot1 Euclidean step2 second coefficient multiple = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore merge original slot1 Euclidean step2 second coefficient difference := sub(0, 3) After merge original slot1 Euclidean step2 second coefficient difference = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · impliesBefore merge original slot1:step=2;current remainders=[3, 1];coefficient pairs=[[1, 0], [-12, 1]];exact ratio=3;floor q=3 After next remainder=0;next signed coefficient pair=[37, -3];a*x+b*y equals next remainder Conditions: none Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse. Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S085 S087 · calculationBefore merge original slot1 final first Bezout term := mul(3, -12) After merge original slot1 final first Bezout term = -36 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore merge original slot1 final second Bezout term := mul(37, 1) After merge original slot1 final second Bezout term = 37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore merge original slot1 final Bezout identity := sum(-36, 37) After merge original slot1 final Bezout identity = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore merge slot1 new residue minus old class := sub(6, 0) After merge slot1 new residue minus old class = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · calculationBefore merge slot1 difference divided by gcd := div(6, 1) After merge slot1 difference divided by gcd = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090 S092 · calculationBefore merge slot1 reduced old modulus := div(3, 1) After merge slot1 reduced old modulus = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore merge slot1 reduced new modulus := div(37, 1) After merge slot1 reduced new modulus = 37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · calculationBefore merge slot1 new LCM period := mul(3, 37) After merge slot1 new LCM period = 111 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · calculationBefore merge slot1 reduced-unit inverse exact quotient := div(-12, 37) After merge slot1 reduced-unit inverse exact quotient = -12/37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094 S096 · calculationBefore merge slot1 reduced-unit inverse floor quotient multiple := mul(-1, 37) After merge slot1 reduced-unit inverse floor quotient multiple = -37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore merge slot1 reduced-unit inverse normalized remainder := sub(-12, -37) After merge slot1 reduced-unit inverse normalized remainder = 25 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · impliesBefore merge slot1 reduced-unit inverse:integer value=-12;positive modulus=37;exact ratio=-12/37;floor quotient=-1 After value=q*modulus+remainder;normalized residue=25 in[0,37) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S097 S099 · calculationBefore merge slot1 parameter congruence product := mul(6, 25) After merge slot1 parameter congruence product = 150 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S098 S100 · calculationBefore merge slot1 normalized parameter exact quotient := div(150, 37) After merge slot1 normalized parameter exact quotient = 150/37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · calculationBefore merge slot1 normalized parameter floor quotient multiple := mul(4, 37) After merge slot1 normalized parameter floor quotient multiple = 148 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · calculationBefore merge slot1 normalized parameter normalized remainder := sub(150, 148) After merge slot1 normalized parameter normalized remainder = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · impliesBefore merge slot1 normalized parameter:integer value=150;positive modulus=37;exact ratio=150/37;floor quotient=4 After value=q*modulus+remainder;normalized residue=2 in[0,37) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S102 S104 · calculationBefore merge slot1 old modulus times parameter := mul(3, 2) After merge slot1 old modulus times parameter = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore merge slot1 unnormalized common solution := sum(0, 6) After merge slot1 unnormalized common solution = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore merge slot1 least class representative exact quotient := div(6, 111) After merge slot1 least class representative exact quotient = 2/37 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · calculationBefore merge slot1 least class representative floor quotient multiple := mul(0, 111) After merge slot1 least class representative floor quotient multiple = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore merge slot1 least class representative normalized remainder := sub(6, 0) After merge slot1 least class representative normalized remainder = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · impliesBefore merge slot1 least class representative:integer value=6;positive modulus=111;exact ratio=2/37;floor quotient=0 After value=q*modulus+remainder;normalized residue=6 in[0,111) Conditions: none Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0. Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S108 S110 · impliesBefore old FULL solution class=0 mod3;new original constraint=6 mod37;gcd=1;Bezout coefficients=[-12, 1];reduced coprime moduli=[3, 37] After write old x=a+M*t;reduced equation gives t=2 mod37;ALL merged solutions x=6 mod111 Conditions: none Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point. Step identifiers and dependenciesRule: crt_constructive_merge_all_solutions_not_one_witness · Uses: G001, S109 ## 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, 37], [0, 6], 6, 111, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3de10ce8b06d3e99162908d83e0839e12c3042c33e778260dc54d45a2dd419bc - candidate record SHA-256: 17a972e937c9904eefa1edad892e10a5f3e07c2cb77e725ebb3d0b9ffeab3ffb - semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc - question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0022.json)