MathNet 2015-part-05 — 191 problems (EN–VI) Question 109

Let $a$, $b$, $c$, $d$ be positive integers such that $a c + b d$ is divisible by $a^{2} + b^{2}$. Prove that $\operatorname{gcd}\left(c^{2} + d^{2}, a^{2} + b^{2}\right) \gt 1$.