AMC 12 2011b Problem 19

A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$?

\textbf{(A)}\ \frac{51}{101} \textbf{(B)}\ \frac{50}{99} \textbf{(C)}\ \frac{51}{100} \textbf{(D)}\ \frac{52}{101} \textbf{(E)}\ \frac{13}{25}