# Question 8993: Both Parenthesizations of a Three-Input GCD

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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.

R0016-08993 · q-20261013-advance-20261018-r03-gcd-coprime-0093



## Original question and givens

Natural inputs a=29, b=52, c=81, 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=29, b=52, c=81, 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.


- [mko-gcd-coprime](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/)





## Step-by-step derivation

S001 · impliesBefore

natural triple=[29, 52, 81];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 dependenciesRule: gcd_two_routes_scope · Uses: G001





S002 · impliesBefore

left inner gcd(a,b) Euclid row0:divide natural 29 by positive 52

After

29=0·52+29

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S001





S003 · impliesBefore

phase=left inner gcd(a,b);pair=[29, 52];quotient=0;remainder=29

After

next pair=[52, 29];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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S002





S004 · impliesBefore

left inner gcd(a,b) Euclid row1:divide natural 52 by positive 29

After

52=1·29+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S003





S005 · impliesBefore

phase=left inner gcd(a,b);pair=[52, 29];quotient=1;remainder=23

After

next pair=[29, 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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S004





S006 · impliesBefore

left inner gcd(a,b) Euclid row2:divide natural 29 by positive 23

After

29=1·23+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S005





S007 · impliesBefore

phase=left inner gcd(a,b);pair=[29, 23];quotient=1;remainder=6

After

next pair=[23, 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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S006





S008 · impliesBefore

left inner gcd(a,b) Euclid row3:divide natural 23 by positive 6

After

23=3·6+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S007





S009 · impliesBefore

phase=left inner gcd(a,b);pair=[23, 6];quotient=3;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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S008





S010 · impliesBefore

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S009





S011 · impliesBefore

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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S010





S012 · impliesBefore

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S011





S013 · impliesBefore

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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S012





S014 · impliesBefore

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 dependenciesRule: gcd_terminal_zero_convention · Uses: G001, S013





S015 · impliesBefore

left outer gcd(ab,c) Euclid row0:divide natural 1 by positive 81

After

1=0·81+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S014





S016 · impliesBefore

phase=left outer gcd(ab,c);pair=[1, 81];quotient=0;remainder=1

After

next pair=[81, 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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S015





S017 · impliesBefore

left outer gcd(ab,c) Euclid row1:divide natural 81 by positive 1

After

81=81·1+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S016





S018 · impliesBefore

phase=left outer gcd(ab,c);pair=[81, 1];quotient=81;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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S017





S019 · impliesBefore

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 dependenciesRule: gcd_terminal_zero_convention · Uses: G001, S018





S020 · impliesBefore

right inner gcd(b,c) Euclid row0:divide natural 52 by positive 81

After

52=0·81+52

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S019





S021 · impliesBefore

phase=right inner gcd(b,c);pair=[52, 81];quotient=0;remainder=52

After

next pair=[81, 52];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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S020





S022 · impliesBefore

right inner gcd(b,c) Euclid row1:divide natural 81 by positive 52

After

81=1·52+29

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S021





S023 · impliesBefore

phase=right inner gcd(b,c);pair=[81, 52];quotient=1;remainder=29

After

next pair=[52, 29];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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S022





S024 · impliesBefore

right inner gcd(b,c) Euclid row2:divide natural 52 by positive 29

After

52=1·29+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S023





S025 · impliesBefore

phase=right inner gcd(b,c);pair=[52, 29];quotient=1;remainder=23

After

next pair=[29, 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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S024





S026 · impliesBefore

right inner gcd(b,c) Euclid row3:divide natural 29 by positive 23

After

29=1·23+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S025





S027 · impliesBefore

phase=right inner gcd(b,c);pair=[29, 23];quotient=1;remainder=6

After

next pair=[23, 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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S026





S028 · impliesBefore

right inner gcd(b,c) Euclid row4:divide natural 23 by positive 6

After

23=3·6+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S027





S029 · impliesBefore

phase=right inner gcd(b,c);pair=[23, 6];quotient=3;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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S028





S030 · impliesBefore

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S029





S031 · impliesBefore

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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S030





S032 · impliesBefore

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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S031





S033 · impliesBefore

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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S032





S034 · impliesBefore

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 dependenciesRule: gcd_terminal_zero_convention · Uses: G001, S033





S035 · impliesBefore

right outer gcd(a,bc) Euclid row0:divide natural 29 by positive 1

After

29=29·1+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 dependenciesRule: natural_euclidean_division_witness · Uses: G001, S034





S036 · impliesBefore

phase=right outer gcd(a,bc);pair=[29, 1];quotient=29;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 dependenciesRule: gcd_common_divisor_invariance · Uses: G001, S035





S037 · impliesBefore

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 dependenciesRule: gcd_terminal_zero_convention · Uses: G001, S036





S038 · impliesBefore

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 dependenciesRule: 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

(29, 52, 81, 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: 1fa31a2769ca89ff598d8d11b3e9e156c2db7d9b49e2e9d61c44fc0d9fd6d882
- candidate record SHA-256: 062fc3d774ee176a052049c7e39be7d4c21704b33a815b6bec91ecc0b941a559
- semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
- question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r03-gcd-coprime-0093.json)