First, I do NOT want the answer to the question, I just need to make sure that there is not a mistake in the wording.
The question states:
In particular, I think the $p^{k-1}(p-1)$ is wrong.
There are $(p-1)$ non-zero equivalent classes in this problem, each one ocurring in between two consecutive powers of $p$. There are $k$ such intervals, so shouldn't there be $k(p-1)$ such integers?
The wording is correct.
The p-1 consecutive values are not between consecutive powers of p but between consecutive multiples of p.