Problem

Source: Romania JBMO tst 2023 day2 p4

Tags: number theory



Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that $n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$ (where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).