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R0016-08988 · q-20261013-advance-20261018-r03-gcd-coprime-0088
Natural inputs a=24, b=42, c=66, 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=24, b=42, c=66, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).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.
Before
natural triple=[24, 42, 66];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.
Rule: gcd_two_routes_scope · Uses: G001
Before
left inner gcd(a,b) Euclid row0:divide natural 24 by positive 42
After
24=0·42+24
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S001
Before
phase=left inner gcd(a,b);pair=[24, 42];quotient=0;remainder=24
After
next pair=[42, 24];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.
Rule: gcd_common_divisor_invariance · Uses: G001, S002
Before
left inner gcd(a,b) Euclid row1:divide natural 42 by positive 24
After
42=1·24+18
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S003
Before
phase=left inner gcd(a,b);pair=[42, 24];quotient=1;remainder=18
After
next pair=[24, 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.
Rule: gcd_common_divisor_invariance · Uses: G001, S004
Before
left inner gcd(a,b) Euclid row2:divide natural 24 by positive 18
After
24=1·18+6
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S005
Before
phase=left inner gcd(a,b);pair=[24, 18];quotient=1;remainder=6
After
next pair=[18, 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.
Rule: gcd_common_divisor_invariance · Uses: G001, S006
Before
left inner gcd(a,b) Euclid row3:divide natural 18 by positive 6
After
18=3·6+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S007
Before
phase=left inner gcd(a,b);pair=[18, 6];quotient=3;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.
Rule: gcd_common_divisor_invariance · Uses: G001, S008
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.
Rule: gcd_terminal_zero_convention · Uses: G001, S009
Before
left outer gcd(ab,c) Euclid row0:divide natural 6 by positive 66
After
6=0·66+6
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S010
Before
phase=left outer gcd(ab,c);pair=[6, 66];quotient=0;remainder=6
After
next pair=[66, 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.
Rule: gcd_common_divisor_invariance · Uses: G001, S011
Before
left outer gcd(ab,c) Euclid row1:divide natural 66 by positive 6
After
66=11·6+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S012
Before
phase=left outer gcd(ab,c);pair=[66, 6];quotient=11;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.
Rule: gcd_common_divisor_invariance · Uses: G001, S013
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.
Rule: gcd_terminal_zero_convention · Uses: G001, S014
Before
right inner gcd(b,c) Euclid row0:divide natural 42 by positive 66
After
42=0·66+42
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S015
Before
phase=right inner gcd(b,c);pair=[42, 66];quotient=0;remainder=42
After
next pair=[66, 42];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.
Rule: gcd_common_divisor_invariance · Uses: G001, S016
Before
right inner gcd(b,c) Euclid row1:divide natural 66 by positive 42
After
66=1·42+24
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S017
Before
phase=right inner gcd(b,c);pair=[66, 42];quotient=1;remainder=24
After
next pair=[42, 24];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.
Rule: gcd_common_divisor_invariance · Uses: G001, S018
Before
right inner gcd(b,c) Euclid row2:divide natural 42 by positive 24
After
42=1·24+18
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S019
Before
phase=right inner gcd(b,c);pair=[42, 24];quotient=1;remainder=18
After
next pair=[24, 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.
Rule: gcd_common_divisor_invariance · Uses: G001, S020
Before
right inner gcd(b,c) Euclid row3:divide natural 24 by positive 18
After
24=1·18+6
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S021
Before
phase=right inner gcd(b,c);pair=[24, 18];quotient=1;remainder=6
After
next pair=[18, 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.
Rule: gcd_common_divisor_invariance · Uses: G001, S022
Before
right inner gcd(b,c) Euclid row4:divide natural 18 by positive 6
After
18=3·6+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S023
Before
phase=right inner gcd(b,c);pair=[18, 6];quotient=3;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.
Rule: gcd_common_divisor_invariance · Uses: G001, S024
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.
Rule: gcd_terminal_zero_convention · Uses: G001, S025
Before
right outer gcd(a,bc) Euclid row0:divide natural 24 by positive 6
After
24=4·6+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S026
Before
phase=right outer gcd(a,bc);pair=[24, 6];quotient=4;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.
Rule: gcd_common_divisor_invariance · Uses: G001, S027
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.
Rule: gcd_terminal_zero_convention · Uses: G001, S028
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.
Rule: gcd_parenthesizations_actual_comparison · Uses: G001, S029
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.(24, 42, 66, 6, 6, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ff1d7c7287e6a1fd5e8385a100078315f6a205a6b838a7b7e787553bcdc3ea93candidate record SHA-256: 1c372b5ae426a698001274ca11d94dc704b97805b9a0a7920635eabe22e7c86bsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED