Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it Ukraine

Find all triples of positive integers $a, b, c$ , such that
$a + [a, b] = b + [b, c] = c + [c, a],$
where by $[x, y]$ we denote the smallest common multiple of $x, y$ .

Solution

Note that $b$ , $[a, b]$ and $[b, c]$ are divisible by $b$ , so $a$ is divisible by $b$ . Similarly, $c$ is divisible by $a$ , and $b$ is divisible by $c$ , so $a = b = c$ .

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