2021 AMC 10B Problem 25

Let S be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30, inclusive. Exactly 300 points in S lie on or below a line with equation y = mx. The possible values of m lie in an interval of length \frac{a}{b}, where a and b are relatively prime positive integers. What is a + b ?

\textbf{(A)} ~31 \qquad\textbf{(B)} ~47 \qquad\textbf{(C)} ~62 \qquad\textbf{(D)} ~72 \qquad\textbf{(E)} ~85

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.