Solving for x in flooring functions

Question

Say you have the equation:

$x\lfloor{\frac{999}{x}}\rfloor = 999$

where x ∈ { 15, 16, 17, ..., 35 }

How would you go about solving for $x$?

Answer 1

Well, assuming $x\neq 0$, this directly implies $$ \left\lfloor\frac{999}{x}\right\rfloor=\frac{999}{x}. $$

So, the question becomes: for what numbers $w$ is it true that $\lfloor w\rfloor=w$?