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Question 31794: Exact Euclidean Remainder Trace and Natural GCD

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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← Question derivation index · Original question batch

R0062-31794 · q-20261013-advance-20261018-r03-gcd-coprime-0019

Original question and givens

Use natural gcd with a=161, b=102. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.

  • G001 — Use natural gcd with a=161, b=102. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.

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-ext18-gcd-euclid-trace, Nat_gcd, gcd-euclid-trace, no-form-field]

After

derived = [[a, b, ordered_Euclid_rows, gcd]]

Conditions: none

Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.

Step identifiers and dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · calculation

Before

GCD Euclid row0 ORIGINAL quotient times y := mul(1, 102)

After

GCD Euclid row0 ORIGINAL quotient times y = 102

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

GCD Euclid row0 ORIGINAL x minus qy := sub(161, 102)

After

GCD Euclid row0 ORIGINAL x minus qy = 59

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [161, 102, 1]

After

derived = [102, 59, [161, 102, 59]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S003

S005 · calculation

Before

GCD Euclid row1 ORIGINAL quotient times y := mul(1, 59)

After

GCD Euclid row1 ORIGINAL quotient times y = 59

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

GCD Euclid row1 ORIGINAL x minus qy := sub(102, 59)

After

GCD Euclid row1 ORIGINAL x minus qy = 43

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [102, 59, 1]

After

derived = [59, 43, [102, 59, 43]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S006

S008 · calculation

Before

GCD Euclid row2 ORIGINAL quotient times y := mul(1, 43)

After

GCD Euclid row2 ORIGINAL quotient times y = 43

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

GCD Euclid row2 ORIGINAL x minus qy := sub(59, 43)

After

GCD Euclid row2 ORIGINAL x minus qy = 16

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [59, 43, 1]

After

derived = [43, 16, [59, 43, 16]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S009

S011 · calculation

Before

GCD Euclid row3 ORIGINAL quotient times y := mul(2, 16)

After

GCD Euclid row3 ORIGINAL quotient times y = 32

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

GCD Euclid row3 ORIGINAL x minus qy := sub(43, 32)

After

GCD Euclid row3 ORIGINAL x minus qy = 11

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [43, 16, 2]

After

derived = [32, 11, [43, 16, 11]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S012

S014 · calculation

Before

GCD Euclid row4 ORIGINAL quotient times y := mul(1, 11)

After

GCD Euclid row4 ORIGINAL quotient times y = 11

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

GCD Euclid row4 ORIGINAL x minus qy := sub(16, 11)

After

GCD Euclid row4 ORIGINAL x minus qy = 5

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [16, 11, 1]

After

derived = [11, 5, [16, 11, 5]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S015

S017 · calculation

Before

GCD Euclid row5 ORIGINAL quotient times y := mul(2, 5)

After

GCD Euclid row5 ORIGINAL quotient times y = 10

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

GCD Euclid row5 ORIGINAL x minus qy := sub(11, 10)

After

GCD Euclid row5 ORIGINAL x minus qy = 1

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [11, 5, 2]

After

derived = [10, 1, [11, 5, 1]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S018

S020 · calculation

Before

GCD Euclid row6 ORIGINAL quotient times y := mul(5, 1)

After

GCD Euclid row6 ORIGINAL quotient times y = 5

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

GCD Euclid row6 ORIGINAL x minus qy := sub(5, 5)

After

GCD Euclid row6 ORIGINAL x minus qy = 0

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [5, 1, 5]

After

derived = [5, 0, [5, 1, 0]]

Conditions: none

With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.

Step identifiers and dependencies

Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S021

S023 · implies

Before

source roles = [161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]]]

After

derived = [1, 0]

Conditions: none

At second value0 retain the last x.For nonzero x,gcd(x,0)=x;the two-way common-divisor invariant transfers this to the original pair.At(0,0)use the EXPLICIT convention gcd=0,not a claim that0 is the largest positive divisor.

Step identifiers and dependencies

Rule: GCD_TERMINAL_last_x_ZEROsecond_and_explicit_gcd00_convention · Uses: G001, S022

Reverse checks

  • S023 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S022 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Recompute GCD Euclid row6 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute GCD Euclid row6 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Recompute GCD Euclid row5 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute GCD Euclid row5 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Recompute GCD Euclid row4 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute GCD Euclid row4 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Recompute GCD Euclid row3 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute GCD Euclid row3 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Recompute GCD Euclid row2 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute GCD Euclid row2 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Recompute GCD Euclid row1 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute GCD Euclid row1 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Recompute GCD Euclid row0 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute GCD Euclid row0 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

(161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 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: 545b2cceab3ded845068912890776ca95bd5ae2049236343d2e3746e30f27009
  • candidate record SHA-256: 632670a5738e00c0108522ec97cca2f3167e95a106ebc65fc744548ece68bec5
  • semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
  • question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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