MathNet 2016-part-02 — 184 problems (EN–VI) Question 155

Find all positive integer numbers $a, b, c$ such that $a+b = (a,b)^2$, $b+c = (b,c)^2$, $c+a = (c,a)^2$. Here denoted by $(a,b)$ the greatest common divisor of $a, b$.