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Question 34819: Reduced Rational Form

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0068-34819 · q-20260920-rational-reduced-form-0019

Original question and givens

Reduce 171/12 to an exact lowest-terms form with a positive denominator.

  • G001 — Reduce 171/12 to an exact lowest-terms form with a positive denominator.

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL rational reduction, 171, 12]

After

derived = [14, 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.

Step identifiers and dependencies

Rule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S001

S003 · implies

Before

source roles = [ORIGINAL rational reduction, 12, 3]

After

derived = [4, 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.

Step identifiers and dependencies

Rule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S002

S004 · calculation

Before

Rational reduction ORIGINAL sign-correct numerator divided by gcd := div(171, 3)

After

Rational reduction ORIGINAL sign-correct numerator divided by gcd = 57

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [171, 12, 3]

After

derived = [57, 4, 57/4]

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.

Step identifiers and dependencies

Rule: Rational_REDUCE_both_components_same_NONzero_gcd · Uses: G001, S005

Reverse checks

  • S006 · 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.
  • S005 · premise_and_result_audit — Recompute Rational reduction ORIGINAL positive denominator divided by gcd with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Rational reduction ORIGINAL sign-correct numerator divided by gcd with exact rational arithmetic, not rounded decimals.
  • 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.

Result

57/4

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: af7fb4c46652039682572baa94d9b4e730ad0be33566f0577e42fa0d6344a97d
  • candidate record SHA-256: 14fe3d4adfc36221b6c7eb6db7c220b0689aa36d7db7ddb6602a520dbf3056d0
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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