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R0068-34825 · q-20260920-rational-reduced-form-0025
Reduce 225/12 to an exact lowest-terms form with a positive denominator.
G001 — Reduce 225/12 to an exact lowest-terms form with a positive denominator.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.
Before
source roles = [qf-rational-reduced-form, [p, q]]
After
derived = [BASE-no-required-operation, no-invented-domain]
Conditions: none
Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact.
Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001
Before
source roles = [ORIGINAL rational reduction, 225, 12]
After
derived = [18, 9]
Conditions: none
Use a=b*q+r with0≤r<b,preserving gcd(a,b)=gcd(b,r).Remainders strictly decrease,so the process terminates.
Rule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S001
Before
source roles = [ORIGINAL rational reduction, 12, 9]
After
derived = [1, 3]
Conditions: none
Use a=b*q+r with0≤r<b,preserving gcd(a,b)=gcd(b,r).Remainders strictly decrease,so the process terminates.
Rule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S002
Before
source roles = [ORIGINAL rational reduction, 9, 3]
After
derived = [3, 0]
Conditions: none
Use a=b*q+r with0≤r<b,preserving gcd(a,b)=gcd(b,r).Remainders strictly decrease,so the process terminates.
Rule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S003
Before
Rational reduction ORIGINAL sign-correct numerator divided by gcd := div(225, 3)
After
Rational reduction ORIGINAL sign-correct numerator divided by gcd = 75
Conditions: none
Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Rational reduction ORIGINAL positive denominator divided by gcd := div(12, 3)
After
Rational reduction ORIGINAL positive denominator divided by gcd = 4
Conditions: none
Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [225, 12, 3]
After
derived = [75, 4, 75/4]
Conditions: none
Divide BOTH components by the SAME positive gcd and normalize the denominator sign,preserving value and yielding coprime components.RETAIN/1 for integers.This differs from sign-only normalization;preserve the ORIGINAL raw p/q.
Rule: Rational_REDUCE_both_components_same_NONzero_gcd · Uses: G001, S006
S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S006 · premise_and_result_audit — Recompute Rational reduction ORIGINAL positive denominator divided by gcd with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Rational reduction ORIGINAL sign-correct numerator divided by gcd with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.75/4
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 37a5a9f61fbc33857ac81fd4282913b7357758482571a13cc924b17105230c2fcandidate record SHA-256: 07b9205d31af16957e41cc9d27b41b15d8e2576b52fddda6ff64b533a8879378semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066edquestion-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229Lean: NOT_RUN · human review: NOT_PERFORMED