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Question 32151: Fraction Reduction and Normalization

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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R0063-32151 · q-20260930-r01-fraction-reduction-0001

Original question and givens

Reduce the valid integer fraction -24/(2) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

  • G001 — Reduce the valid integer fraction -24/(2) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

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-fraction-reduction, reduction, no-required-domain, []]

After

derived = [[reduced_fraction_RETAIN1_STRING]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [-24, 2, []]

After

derived = [Yes]

Conditions: none

Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero.

Step identifiers and dependencies

Rule: Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity · Uses: G001, S001

S003 · implies

Before

source roles = [Fraction reduction, -24, 2]

After

derived = [-24, 2]

Conditions: none

A denominator must FIRST be nonzero.If negative,negate numerator AND denominator together.Derived numerator/denominator are not RAW-SAFE recapped;changing only the denominator would change the value.

Step identifiers and dependencies

Rule: Fraction_DEN_nonzero_sign_normalized_BEFORE_gcd · Uses: G001, S002

S004 · calculation

Before

Fraction reduction gcd step0 ORIGINAL quotient times divisor := mul(12, 2)

After

Fraction reduction gcd step0 ORIGINAL quotient times divisor = 24

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Fraction reduction gcd step0 ORIGINAL dividend minus product := sub(24, 24)

After

Fraction reduction gcd step0 ORIGINAL dividend minus product = 0

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [Fraction reduction, 0, 24, 2, 12]

After

derived = [24, 0]

Conditions: none

Use x=qy+r,0≤r<y,to replace gcd(x,y)by gcd(y,r),preserving common divisors BOTH ways.Divide ONLY by nonzero y;remainders strictly decrease to0.

Step identifiers and dependencies

Rule: Fraction_EUCLID_all_remainders_and_common_divisor_invariant · Uses: G001, S005

S007 · calculation

Before

Fraction reduction ORIGINAL numerator over positive gcd := div(-24, 2)

After

Fraction reduction ORIGINAL numerator over positive gcd = -12

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Fraction reduction ORIGINAL denominator over positive gcd := div(2, 2)

After

Fraction reduction ORIGINAL denominator over positive gcd = 1

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Fraction reduction reverse reduced numerator times gcd := mul(-12, 2)

After

Fraction reduction reverse reduced numerator times gcd = -24

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Fraction reduction reverse reduced denominator times gcd := mul(1, 2)

After

Fraction reduction reverse reduced denominator times gcd = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [Fraction reduction, -24, 2, 2]

After

derived = [-12, 1, -24, 2, -12/1]

Conditions: none

Divide BOTH components by the positive gcd,retaining value and a positive lowest-terms denominator.Multiply back to recover both components.Zero is0/1;integers RETAIN/1,not decimals.

Step identifiers and dependencies

Rule: Fraction_REDUCED_positive_den_ZERO0over1_RETAIN_integer1 · Uses: G001, S010

Reverse checks

  • S011 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Recompute Fraction reduction reverse reduced denominator times gcd with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Fraction reduction reverse reduced numerator times gcd with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL denominator over positive gcd with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL numerator over positive gcd with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Recompute Fraction reduction gcd step0 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Fraction reduction gcd step0 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-12/1

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: 5d2a229f527b21d615f62e3fa831765d520ac5429e29aae44063140050df4eef
  • candidate record SHA-256: 0acaeb55542cb8ecd3d3d14089db4fe2a664fe9ee49785238ec1746670be8b71
  • semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
  • question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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