Question 42
For a given positive integer $k$ let $S(k)$ denote the sum of all numbers from the set $\{1, 2, \dots, k\}$ relatively prime to $k$. Let $m$ be a positive integer and $n$ an odd positive integer. Prove that there exist positive integers $x$ and $y$ such that $m$ divides $x$ and $2S(x) = y^n$.