Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$. Here $(a, b)$ denotes the greatest common divisor of integers $a, b$. (Proposed by Mykhailo Shtandenko)
Source: Kyiv City MO 2022 Round 2, Problem 9.1
Tags: number theory, greatest common divisor
Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$. Here $(a, b)$ denotes the greatest common divisor of integers $a, b$. (Proposed by Mykhailo Shtandenko)