If gcd(x, y) Divides (x + y) Then There Are Infinite Values Of x And y

Let's say there exists an integer $z$, such that:

Now let $x+y=s$ and $(x,y)=d$, then:

This means:

Since $z$ is some integer, then there are infinite possible values of $m$ and $n$, which means there will also be infinite possible values for $x$ and $y$. If $z$ wasn't an integer, then both $m$ and $n$ wouldn't be integers as well, which means no integers $x$ and $y$ would exist.