smc

It is very easy to see that the $+2$ in the graph does not impact whether it passes through the lattice. We need to make sure that $m$ cannot be in the form of $\frac{a}{b}$ for $1\le b\le 100$. Otherwise, the graph $y=mx$ passes through the lattice point at $x=b$. We only need to worry about $\frac{a}{b}$ very close to $\frac{1}{2}$, $\frac{n+1}{2n+1}$, $\frac{n+1}{2n}$ will be the only case we need to worry about and we want the minimum of those, clearly for $1\le b\le 100$, the smallest is $\frac{50}{99}$, so answer is $\boxed{\frac{50}{99}}$ (In other words we are trying to find the smallest $m>\frac{1}{2}$ such that $b\le 100$.)