imc

For $y=mx+2$ to not pass through any lattice points with $0<x\le 100$ 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 $t\le 100$. 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 $1,\frac{2}{3},\frac{3}{4},\frac{3}{5},\dots ,\frac{50}{98},\frac{50}{99},\frac{51}{100}$ respectively, and the smallest of these is $\boxed{\frac{50}{99}}$.