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Question 8987: Both Parenthesizations of a Three-Input 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

R0016-08987 · q-20261013-advance-20261018-r03-gcd-coprime-0087

Original question and givens

Natural inputs a=23, b=40, c=63, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).

  • G001 — Natural inputs a=23, b=40, c=63, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).

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

natural triple=[23, 40, 63];gcd(0,0)=0

After

compute both complete parenthesizations

Conditions: none

Compute both actual Euclidean routes rather than repeating the associativity conclusion.Inputs may include0.

Step identifiers and dependencies

Rule: gcd_two_routes_scope · Uses: G001

S002 · implies

Before

left inner gcd(a,b) Euclid row0:divide natural 23 by positive 40

After

23=0·40+23

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S001

S003 · implies

Before

phase=left inner gcd(a,b);pair=[23, 40];quotient=0;remainder=23

After

next pair=[40, 23];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S002

S004 · implies

Before

left inner gcd(a,b) Euclid row1:divide natural 40 by positive 23

After

40=1·23+17

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S003

S005 · implies

Before

phase=left inner gcd(a,b);pair=[40, 23];quotient=1;remainder=17

After

next pair=[23, 17];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S004

S006 · implies

Before

left inner gcd(a,b) Euclid row2:divide natural 23 by positive 17

After

23=1·17+6

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S005

S007 · implies

Before

phase=left inner gcd(a,b);pair=[23, 17];quotient=1;remainder=6

After

next pair=[17, 6];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S006

S008 · implies

Before

left inner gcd(a,b) Euclid row3:divide natural 17 by positive 6

After

17=2·6+5

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S007

S009 · implies

Before

phase=left inner gcd(a,b);pair=[17, 6];quotient=2;remainder=5

After

next pair=[6, 5];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S008

S010 · implies

Before

left inner gcd(a,b) Euclid row4:divide natural 6 by positive 5

After

6=1·5+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S009

S011 · implies

Before

phase=left inner gcd(a,b);pair=[6, 5];quotient=1;remainder=1

After

next pair=[5, 1];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S010

S012 · implies

Before

left inner gcd(a,b) Euclid row5:divide natural 5 by positive 1

After

5=5·1+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S011

S013 · implies

Before

phase=left inner gcd(a,b);pair=[5, 1];quotient=5;remainder=0

After

next pair=[1, 0];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S012

S014 · implies

Before

phase=left inner gcd(a,b);terminal pair=[1, 0]

After

gcd=1

Conditions: none

At termination gcd(x,0)=x,including the natural convention gcd(0,0)=0.Never take a remainder modulo0,and do not infer that both inputs are prime.

Step identifiers and dependencies

Rule: gcd_terminal_zero_convention · Uses: G001, S013

S015 · implies

Before

left outer gcd(ab,c) Euclid row0:divide natural 1 by positive 63

After

1=0·63+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S014

S016 · implies

Before

phase=left outer gcd(ab,c);pair=[1, 63];quotient=0;remainder=1

After

next pair=[63, 1];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S015

S017 · implies

Before

left outer gcd(ab,c) Euclid row1:divide natural 63 by positive 1

After

63=63·1+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S016

S018 · implies

Before

phase=left outer gcd(ab,c);pair=[63, 1];quotient=63;remainder=0

After

next pair=[1, 0];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S017

S019 · implies

Before

phase=left outer gcd(ab,c);terminal pair=[1, 0]

After

gcd=1

Conditions: none

At termination gcd(x,0)=x,including the natural convention gcd(0,0)=0.Never take a remainder modulo0,and do not infer that both inputs are prime.

Step identifiers and dependencies

Rule: gcd_terminal_zero_convention · Uses: G001, S018

S020 · implies

Before

right inner gcd(b,c) Euclid row0:divide natural 40 by positive 63

After

40=0·63+40

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S019

S021 · implies

Before

phase=right inner gcd(b,c);pair=[40, 63];quotient=0;remainder=40

After

next pair=[63, 40];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S020

S022 · implies

Before

right inner gcd(b,c) Euclid row1:divide natural 63 by positive 40

After

63=1·40+23

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S021

S023 · implies

Before

phase=right inner gcd(b,c);pair=[63, 40];quotient=1;remainder=23

After

next pair=[40, 23];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S022

S024 · implies

Before

right inner gcd(b,c) Euclid row2:divide natural 40 by positive 23

After

40=1·23+17

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S023

S025 · implies

Before

phase=right inner gcd(b,c);pair=[40, 23];quotient=1;remainder=17

After

next pair=[23, 17];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S024

S026 · implies

Before

right inner gcd(b,c) Euclid row3:divide natural 23 by positive 17

After

23=1·17+6

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S025

S027 · implies

Before

phase=right inner gcd(b,c);pair=[23, 17];quotient=1;remainder=6

After

next pair=[17, 6];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S026

S028 · implies

Before

right inner gcd(b,c) Euclid row4:divide natural 17 by positive 6

After

17=2·6+5

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S027

S029 · implies

Before

phase=right inner gcd(b,c);pair=[17, 6];quotient=2;remainder=5

After

next pair=[6, 5];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S028

S030 · implies

Before

right inner gcd(b,c) Euclid row5:divide natural 6 by positive 5

After

6=1·5+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S029

S031 · implies

Before

phase=right inner gcd(b,c);pair=[6, 5];quotient=1;remainder=1

After

next pair=[5, 1];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S030

S032 · implies

Before

right inner gcd(b,c) Euclid row6:divide natural 5 by positive 1

After

5=5·1+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S031

S033 · implies

Before

phase=right inner gcd(b,c);pair=[5, 1];quotient=5;remainder=0

After

next pair=[1, 0];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S032

S034 · implies

Before

phase=right inner gcd(b,c);terminal pair=[1, 0]

After

gcd=1

Conditions: none

At termination gcd(x,0)=x,including the natural convention gcd(0,0)=0.Never take a remainder modulo0,and do not infer that both inputs are prime.

Step identifiers and dependencies

Rule: gcd_terminal_zero_convention · Uses: G001, S033

S035 · implies

Before

right outer gcd(a,bc) Euclid row0:divide natural 23 by positive 1

After

23=23·1+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S034

S036 · implies

Before

phase=right outer gcd(a,bc);pair=[23, 1];quotient=23;remainder=0

After

next pair=[1, 0];0<=remainder<divisor

Conditions: none

From x=q*y+r,a number divides both x,y exactly when it divides both y,r.The remainder is strictly smaller than the positive divisor,so the second argument eventually becomes0.

Step identifiers and dependencies

Rule: gcd_common_divisor_invariance · Uses: G001, S035

S037 · implies

Before

phase=right outer gcd(a,bc);terminal pair=[1, 0]

After

gcd=1

Conditions: none

At termination gcd(x,0)=x,including the natural convention gcd(0,0)=0.Never take a remainder modulo0,and do not infer that both inputs are prime.

Step identifiers and dependencies

Rule: gcd_terminal_zero_convention · Uses: G001, S036

S038 · implies

Before

gcd(a,b)=1;gcd(b,c)=1

After

left=1;right=1;equal=Yes

Conditions: none

Feed each inner result into its own outer route and compare.The general reason is equality of the three-number common-divisor sets,but this instance retains the actual quotient/remainder witnesses.

Step identifiers and dependencies

Rule: gcd_parenthesizations_actual_comparison · Uses: G001, S037

Reverse checks

  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(23, 40, 63, 1, 1, Yes)

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: 56f6b569c99861a2cff3dfeb43310a7692affd35346060c6ae6f2455217bfe7a
  • candidate record SHA-256: 8906168a08888e72398e777b85468c2aabed0ba87d271634281d1704cb67eb2b
  • semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
  • question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON