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Question 34810: 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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R0068-34810 · q-20260920-rational-reduced-form-0010

Original question and givens

Reduce -60/8 to an exact lowest-terms form with a positive denominator.

  • G001 — Reduce -60/8 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, 60, 8]

After

derived = [7, 4]

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, 8, 4]

After

derived = [2, 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(-60, 4)

After

Rational reduction ORIGINAL sign-correct numerator divided by gcd = -15

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(8, 4)

After

Rational reduction ORIGINAL positive denominator divided by gcd = 2

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 = [-60, 8, 4]

After

derived = [-15, 2, -15/2]

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

-15/2

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: 45773f27d02a2482a509356497e1c053509ce42b40b0ef90ffceec763806dd18
  • candidate record SHA-256: fbc5f1399cf7348a77e0442ae17ae4629ec73e3b727a4707172755f1a980f36a
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON