Let $a,b,c\in\mathbb{Z}^+$ and $a<b$. Given $n=ab$ such that $b-a$ is minimum, what condition on $n$ and $c$ guarantees that there is no factor of $nc^2$ between $ac$ and $c\sqrt{n}$?
Example: Suppose $n=480$, then the two factors of $n$ such that their difference is minimum are $a=20$ and $b=24$. Let $c=3$, then the factors of $nc^2=4320$ such that their difference is minimum are $ac=60$ and $bc=72$. However, if $c=6$, then the factors of $nc^2=17280$ such that their difference is minimum are $128$ and $135$ and not $ac=120$ and $bc=144$.
The case is trivial if $n$ is a square number but is there a general condition?