Answer     $4$

Let $a$ and $b$ be the side lengths of a desired rectangle, then we have $ab=2015$. Hence, this problem is equivalent to find the number of pairs of $2015$'s divisors. Because $2015= 5\times 13\times 31$, therefore it has $(1+1)(1+1)(1+1)=8$ distinct divisors which can form $\boxed{4}$ pairs.