AMC 10 2021b 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 $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ $\textbf{(A)} ~31 \textbf{(B)} ~47 \textbf{(C)} ~62 \textbf{(D)} ~72 \textbf{(E)} ~85$