The positive integers $x$ and $y$ are such that the number $x^{2019} + x + y^2$ is divisible by $xy$.

a) Give an example of such integers $x$ and $y, \operatorname{with} x > y$.

b) Verify that, necessarily, $x$ is a perfect square.
Solution

a) It suffices to choose $x=4$ and $y=2$, because $4^{2019}+4+2^{2}=8 \cdot\left(2^{4035}+1\right)$ is divisible by $4 \cdot 2=8$.
b) Let $d=\gcd(x, y)$. Thus, $x=d m$ and $y=d n$, with $\gcd(m, n)=1$. Therefore,

$$
\frac{x^{2019}+x+y^{2}}{x y}=\frac{d^{2018} m^{2019}+m+d n^{2}}{d m n}
$$

is an integer. Since $d$ divides $d^{2018} m^{2019}+m+d n^{2}$, we can conclude that $d$ divides $m$. Moreover, from $\gcd(m, n)=1$ and $\frac{d^{2018} m^{2019}+m+d n^{2}}{m} \in \mathbb{Z}$, we can conclude that $m$ divides $d$. Since $m$ and $d$ are positive, $m=d$ and $x=d m=d^{2}$.