MathNet 2019-part-02 — 184 problems (EN–VI) Question 168

Let $x$ and $y$ be two integers such that $5x + 6y$ and $6x + 5y$ are perfect squares. Prove that $x$ and $y$ are both divisible by $11$.

Note: an integer $n$ is called a perfect square if it is the square of an integer.