So for an iff, I know you must prove it both ways. I have proven the converse by the idea that 

$\frac{a}{b} + \frac{b}{a} = \frac{a}{a} + \frac{a}{a} = 2 $ which is an integer. 

But I am struggling with the direct proof that if $\frac{a}{b} + \frac{b}{a}$ is an integer then $a = b$.

I have gotten to $ab \mid a^2 + b^2$ but I end up trying to justify why $\frac{b^2}{a}$ is not an integer which I cannot even prove intuitively.