The question and its solution are given below:
I am doubting if the solution is correct or no, for my opinion I see that most of it is correct ..... but I do not understand the last part, that is how $$(a) \sum_{d|{p^k}} \sigma(d) = \sum_{i = 0}^k (k +1 - i)p^i$$, also I do not understand how last equality in the other sum is calculated, could anyone explain this for me please?