If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers

If $x+y$ is even, then either both $x$ and $y$ are odd, or they are both even. Therefore $x$ and $y$ are both even because otherwise $xy$ would be odd.

Note that $$ x^2=(x+y)x-xy $$ and $$ y^2=(x+y)y-xy $$ being the differences of two even numbers, are both even.

Thus, both $x$ and $y$ are even.

Observe that $\displaystyle (x+1)(y+1)=xy+x+y+1$ is odd as $xy,x+y$ are even

If $x$ is odd $\iff x+1$ is even