OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 8976: 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

Return to catalogue page 45

← Question derivation index · Original question batch

R0016-08976 · q-20261013-advance-20261018-r03-gcd-coprime-0076

Original question and givens

Natural inputs a=12, b=18, c=30, 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=12, b=18, c=30, 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=[12, 18, 30];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 12 by positive 18

After

12=0·18+12

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=[12, 18];quotient=0;remainder=12

After

next pair=[18, 12];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 18 by positive 12

After

18=1·12+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, S003

S005 · implies

Before

phase=left inner gcd(a,b);pair=[18, 12];quotient=1;remainder=6

After

next pair=[12, 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, S004

S006 · implies

Before

left inner gcd(a,b) Euclid row2:divide natural 12 by positive 6

After

12=2·6+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, S005

S007 · implies

Before

phase=left inner gcd(a,b);pair=[12, 6];quotient=2;remainder=0

After

next pair=[6, 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, S006

S008 · implies

Before

phase=left inner gcd(a,b);terminal pair=[6, 0]

After

gcd=6

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, S007

S009 · implies

Before

left outer gcd(ab,c) Euclid row0:divide natural 6 by positive 30

After

6=0·30+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, S008

S010 · implies

Before

phase=left outer gcd(ab,c);pair=[6, 30];quotient=0;remainder=6

After

next pair=[30, 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, S009

S011 · implies

Before

left outer gcd(ab,c) Euclid row1:divide natural 30 by positive 6

After

30=5·6+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, S010

S012 · implies

Before

phase=left outer gcd(ab,c);pair=[30, 6];quotient=5;remainder=0

After

next pair=[6, 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, S011

S013 · implies

Before

phase=left outer gcd(ab,c);terminal pair=[6, 0]

After

gcd=6

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, S012

S014 · implies

Before

right inner gcd(b,c) Euclid row0:divide natural 18 by positive 30

After

18=0·30+18

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, S013

S015 · implies

Before

phase=right inner gcd(b,c);pair=[18, 30];quotient=0;remainder=18

After

next pair=[30, 18];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, S014

S016 · implies

Before

right inner gcd(b,c) Euclid row1:divide natural 30 by positive 18

After

30=1·18+12

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, S015

S017 · implies

Before

phase=right inner gcd(b,c);pair=[30, 18];quotient=1;remainder=12

After

next pair=[18, 12];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, S016

S018 · implies

Before

right inner gcd(b,c) Euclid row2:divide natural 18 by positive 12

After

18=1·12+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, S017

S019 · implies

Before

phase=right inner gcd(b,c);pair=[18, 12];quotient=1;remainder=6

After

next pair=[12, 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, S018

S020 · implies

Before

right inner gcd(b,c) Euclid row3:divide natural 12 by positive 6

After

12=2·6+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, S019

S021 · implies

Before

phase=right inner gcd(b,c);pair=[12, 6];quotient=2;remainder=0

After

next pair=[6, 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, S020

S022 · implies

Before

phase=right inner gcd(b,c);terminal pair=[6, 0]

After

gcd=6

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, S021

S023 · implies

Before

right outer gcd(a,bc) Euclid row0:divide natural 12 by positive 6

After

12=2·6+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, S022

S024 · implies

Before

phase=right outer gcd(a,bc);pair=[12, 6];quotient=2;remainder=0

After

next pair=[6, 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, S023

S025 · implies

Before

phase=right outer gcd(a,bc);terminal pair=[6, 0]

After

gcd=6

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, S024

S026 · implies

Before

gcd(a,b)=6;gcd(b,c)=6

After

left=6;right=6;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, S025

Reverse checks

  • 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

(12, 18, 30, 6, 6, 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: 664b5d32a460f7d72eebe500e49f4336bda3a99e86c2110b6e2cafdaa1f7a016
  • candidate record SHA-256: cb42af1b1bd6b0a92aff3285f3193bc928b79d62015055e30ae346a21f5f55e8
  • semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
  • question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON