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GCD Condition

Source: 2022 AMC 12B #16

November 17, 2021
number theorygreatest common divisorAMCAMC 12AMC 12 B

Problem Statement

Let a,b,a, b, and cc be positive integers such that a+b+c=23a+b+c=23 and gcd⁡(a,b)+gcd⁡(b,c)+gcd⁡(c,a)=9.\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9. What is the sum of all possible distinct values of a2+b2+c2a^{2}+b^{2}+c^{2}?
<spanclass=′latex−bold′>(A)</span> 259<spanclass=′latex−bold′>(B)</span> 438<spanclass=′latex−bold′>(C)</span> 516<spanclass=′latex−bold′>(D)</span> 625<spanclass=′latex−bold′>(E)</span> 687<span class='latex-bold'>(A)</span> ~259\qquad<span class='latex-bold'>(B)</span> ~438\qquad<span class='latex-bold'>(C)</span> ~516\qquad<span class='latex-bold'>(D)</span> ~625\qquad<span class='latex-bold'>(E)</span> ~687
Proposed by djmathman
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