Problem

Source: 2021 Saudi Arabia BMO TST 1.3

Tags: number theory, divides, divisible



Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.