2200.11 – Positive Integer Solutions: Count Them

The graph is below. The question is: through how many lattice points does the straight line pass?

We could simply tabulate the values of $y = \dfrac{501 - 3x}{5}$ for all integral $x$ using a spreadsheet and count the number of integer $y$ values.

Or, if we use a calculator, we soon see that $y$ is an integer when $x = 2, 7, 12, 17,$ and so on. This suggests that there are two usable values of $x$ in every interval $[0, 10], [11, 20], \ldots [160, 170]$, except in the last one where 167 is not usable. There are 16 such intervals, each with two good $x$'s, plus the extra: $x = 162$. That makes $2 \times 16 + 1 = 33$ good $x$'s, or 33 lattice points. The answer is (a).