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.

Suppose you have some pair of integers $(x_0, y_0)$ that are a solution to $y = \dfrac{4x - 1}{17}$. Then $$\dfrac{4(x_0 + 17) - 1}{17} = \frac{4x_0 + 4 \cdot 17 - 1}{17} = \dfrac{4x_0 -1}{17} + 4$$ so that $\left(x_0 + 17, y_0 + 4 \right)$ is another solution.
Can you find the "smallest" solution, and figure out how many times you can "increment" to another solution with $x, y$ in the proper bounds?
Bonus: Why do we only need to worry about incrementing $x_0$ by multiples of $17$? Why can't there be some solution $(x, y)$ where $x = x_0 + k$, with $1 \le k \le 16$?
