Let $n$ and $k$ be positive integers. Prove that if $n+k$ is a factor of $n^2$ then $k > \sqrt{n}$.
I do not really know how to approach this. I tried letting $(n+k)(n-m) = n^2$ for some positive integer $m$. Using the AP-GP inequality I showed that $k>m$ and I thought to try proof by contradiction but I was not able to progress any further.
Thanks for any help!
Note that $$(n+k)(n-k)=n^2-k^2$$
Since $(n+k)\,|\,n^2$ we see that $(n+k)\,|\,k^2$.
Now, if we had $k≤\sqrt n$ we'd have $k^2≤n<n+k$, a contradiction.