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 $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$?
$\textbf{(A)}\ \frac{51}{101} \qquad\textbf{(B)}\ \frac{50}{99} \qquad\textbf{(C)}\ \frac{51}{100} \qquad\textbf{(D)}\ \frac{52}{101} \qquad\textbf{(E)}\ \frac{13}{25}$
For $y=mx+2$ to not pass through any lattice points with $0<x\leq 100$ is the same as saying that $mx\notin\mathbb Z$ for $x\in\{1,2,\dots,100\}$, or in other words, $m$ is not expressible as a ratio of positive integers $s/t$ with $t\leq 100$. Hence the maximum possible value of $a$ is the first real number after $1/2$ that is so expressible.
For each $d=2,\dots,100$, the smallest multiple of $1/d$ which exceeds $1/2$ is $1,\frac23,\frac34,\frac35,\dots,\frac{50}{98},\frac{50}{99},\frac{51}{100}$ respectively, and the smallest of these is $\boxed{\textbf{(B)}\frac{50}{99}}$.