How can I show that the following number is not divisible by $p$ prime?

Question

Let $p$ be a prime number. Let $k$ be some natural number and $r$ be some nonnegative integer.

Then, I want to show that for $1\leq i\leq p^k-1$,

\begin{equation*} \frac{p^{k+r}m-i}{p^k-i} \end{equation*} is not divisible by $p$, where $m$ is not divisible by $p$.

Can you give me a hint??

Answer 1

For the whole fraction to be divisible by $p$, assuming $i$ is an integer, the numerator must be divisible by $p$. The numerator is only divisible by $p$ when $i$ is divisible by $p$. This just leaves the case $i=ap^n,n<k$ to consider.