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.

We are asked to find integer solutions for $x,y$.
We can get particular solutions simply by plugging in values. Our first solution is obtained as: $y’ = 3$ and $x’ = 13$.
Thus, the general solution for x will be:
$$x = x’ + bn = 13 + 17n$$
Hence lower limit for $n$ is $n \geq 0$.
Now we’ll find upper limit using given restriction $x \leq 500$. Plugging in general solution we’ll get:
$$13 + 17n \leq 500 \Rightarrow 17n \leq 487\Rightarrow n \leq 28$$ 
And our list of possible values of $n$ has following representation:
$$0, 1,2, 3,\cdots 28$$ Also, we can check that for $n=28$, the value of $y$ also lies within $500$.
The total number of elements is thus $\boxed{29}$.
