Find the possible values of $n$?

Question

The equation is $$n=\frac{\sqrt{16m^{2}+x^2}-x}{m}$$ $x>0$,$m>0$ and $n$ is a positive integer.How many values of $n$ are possible?

Answer 1

Hint:

Using this let $4m=2pqk, x=k(p^2-q^2)$

$$n=4\cdot\dfrac{(p^2+q^2)-(p^2-q^2)}{2pq}=\cdots=\dfrac{4q}p$$

So, all we need is $p|4q$