Rewrite the equation as $$n=\frac{2016-20m}{16}=126-\frac{5m}{4}$$

For $n$ to be a positive integer, $4$ must divide $m$ and $126-\frac{5m}{4} > 0 \implies m \le 100$. There are $25$ positive integer $m$ satifying this condition. Therefore the answer to this problem is $25$.