Find the smallest positive integer $m$ such that there exists positive integer $n$ that satisfies... 
Find the smallest positive integer $m$ such that there exists positive integer $n$ that satisfies that $\lvert {n\over m} - {2\over 5} \rvert\le {1\over100}$

I tried to simplify and turn it into the following
$$\lvert {5n-2m\over5m} \rvert\le {1\over100}$$
$$100\cdot\lvert {5n-2m\over5m} \rvert\le1$$
Since $5m>0$ we get that
$$100\cdot\lvert5n-2m\rvert\le5m$$
We can take cases now
$$\begin{cases}
100(5n-2m)\le5m, & \text { if } & 5n-2m\ge0 \\[2ex]
100(2m-5n)\le5m, & \text { if } & 5n-2m\le0
\end{cases}$$
Unfortunately I am unable to continue from here. Any help would be appreciated.

$|\frac{n}{m}\,-\,\frac{2}{5}|\,\leq\frac{1}{100} $
$$ |\frac{100n}{m}\,-\,40|\, \leq 1. $$
39$\leq\frac{100n}{m}\,\leq41$
Therefore $\frac{n}{m}=\frac{2}{5}$ Hence 5n=2m hence min m will be 5 when n=2.
