Find sets of terms of given sequences

Question

I want to find the sets of the terms of the sequences $\left( n-2 \left[\frac{n}{2}\right]\right), \left( n-3 \left[\frac{n}{3}\right]\right) $ and more general, if $m$ is natural the set of the terms of $\left( n-m \left[\frac{n}{m}\right]\right)$.

We know that $\left( n-m \left[\frac{n}{m}\right]\right)$ is equal to the largest integer $k$ for which $k \leq \frac{n}{m}<k+1$.

But how can we find a general form for the terms of the given sequences and consequently the desired sets?

Answer 1

For natural number $m$ and integer $n$, $n-m \left[\frac{n}{m}\right] \equiv n \bmod m$. The sets are those of the integers modulo $m$, $\mathbb{Z}/m\mathbb{Z}$.