If $x, y \leq 500$ then find the number of nonnegative integer solutions to $4 x - 17y = 1$ 
If $x, y \leq 500$ then find the number of nonnegative integer solutions to $4 x - 17y = 1$.

I don't know how to proceed. Please help me out. Thank you.

$$4x-17y=17-16\iff4(x+4)=17(y+1)\iff\dfrac{17(y+1)}4=x+4$$ which is an integer
$\implies4|17(y+1)\implies4|(y+1)$ 
So $y$ can be written as $4m-1$ where $m$ is any integer
$\implies x=17m-4$
We need $0\le17m-4\le500\iff1\le m\le29\ \ \ \  (1)$
and $0\le4m-1\le500\iff1\le m\le125\ \ \ \ (2)$
What is the intersection of $(1),(2)?$
