# Question 34823: Reduced Rational Form

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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.

R0068-34823 · q-20260920-rational-reduced-form-0023



## Original question and givens

Reduce 207/12 to an exact lowest-terms form with a positive denominator.


- G001 — Reduce 207/12 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.


- [mko-rational-number](https://ocme.evemisslab.com/en/mko/mko-rational-number/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001





S002 · impliesBefore

source roles = [ORIGINAL rational reduction, 207, 12]

After

derived = [17, 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 dependenciesRule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S001





S003 · impliesBefore

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 dependenciesRule: Rational_EUCLID_exact_quotient_remainder_preserves_gcd · Uses: G001, S002





S004 · calculationBefore

Rational reduction ORIGINAL sign-correct numerator divided by gcd := div(207, 3)

After

Rational reduction ORIGINAL sign-correct numerator divided by gcd = 69

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [207, 12, 3]

After

derived = [69, 4, 69/4]

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 dependenciesRule: 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

69/4

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: 8cc4919b5c4354cbccbf24e9f2a7fffb31bc7130776411df91077122220ea49b
- candidate record SHA-256: b7ef1a8f97ef1cf6bb2a9e2742ba926b4c5b08f0f8dc512b83a4b3f80de5008e
- semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
- question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260920-rational-reduced-form-0023.json)