If $ \exists x,\lfloor\frac{n}{x}\rfloor=k $, then what's the range of $ x $?($ x, k $ are positive integers)

Question

When I solve some programming problems about Mobius inversion, I have to used it.I think I can let $ n = kt + r $ and use some inequalities, but I can't go on.

Answer 1

$$\eqalign{\left\lfloor\frac nx\right\rfloor=k\quad &\Leftrightarrow\quad k\le\frac nx<k+1\cr &\Leftrightarrow\quad \frac n{k+1}<x\le\frac nk\cr}$$ if $n,k$ are positive.