Strange function

Begin with the equation $xy=2013x+2013y$ . From this we obtain $x(y-2013)=2013y\Rightarrow x=\frac{2013y}{y-2013}=2013+\frac{2013^2}{y-2013}$ . Set $k=y-2013$ ; thus $k$ is an integer. For $x$ to be an integer, we must have that $\frac{2013^2}{k}$ . Any $k$ will determine a $y$ and thus an $x$ , so we only need to find the number of $k$ s. However, this is just the number of divisors of $2013^2=3^211^261^2\Rightarrow(2+1)(2+1)(2+1)=\boxed{27}$ .