I am not sure if the nomenclature is correct but my module says this to be the remainder theorem,
If a natural number $n$ is divided by a natural number $m$ and can be brought in the form: $\frac{x \cdot (a^p)^q}{a^p-1}$ such that $n=x \cdot (a^p)^q$ and $m=a^p-1\,$ where $x,a,p,q \in \mathbb{N}$ and $x \lt m$, then the remainder of the division of $n$ by $m$ is $x$.
This theorem is holds and I have used this in some problems, I was inquisitive about how probably we can prove this? I asked my instructor but according to him I don't need to be bothered about the proof but only concentrate on solving the problems. However I am rather not much convinced; could anybody explain me how we can prove this?