Maximum and minimum values of $\left\lfloor \frac{x}{nm}\right\rfloor - \left\lfloor \frac{1}{m}\left\lfloor \frac{x}{n}\right\rfloor \right\rfloor$

Question

I feel like I need some additional pointers on the following questions as I am unable to come up with a solution for it:

If $m$ and $n$ are any integers, and $x$ is any positive real number, what are the maximum and minimum values of $\lfloor x/nm\rfloor - \lfloor \lfloor x/n\rfloor /m\rfloor$?

Thanks for your help!

Answer 1

It is always $0$: $$\frac{x+mn}{mn}>\left\lfloor\frac x{mn}\right\rfloor\ge\frac x{mn}$$ $$\frac{x+mn}{mn}>\frac{\lfloor x/n\rfloor+m}m>\left\lfloor\frac{\left\lfloor\dfrac x{n}\right\rfloor}m\right\rfloor\ge\frac{\lfloor x/n\rfloor}m\ge \frac x{mn}$$

Answer 2

$\newcommand{\lf}{\left\lfloor}$ $\newcommand{\rf}{\right\rfloor}$

The quantity $\lf x/(mn)\rf=\lf(x/n)/m\rf$ is the number of positive integer multiples of $m$ not exceeding $x/n$, while the quantity $\lf\lf x/n\rf/m\rf$ is the number of positive integer multiples of $m$ not exceeding $\lf x/n\rf$. However, there are no multiples of $m$ (and no integer numbers at all) between $\lf x/n\rf$ and $x/n$! It follows that the two quantities are equal to each other: $$ \lf\frac x{mn}\rf=\lf\frac{\lf\frac{x}{n}\rf}m\rf, $$ for any positive real $x$ and integer $m$ and $n$.