VMO 2021 Problem 4

For an integer $n \geq 2$ , let $s (n)$ be the sum of positive integers not exceeding $n$ and not relatively prime to $n$ . a) Prove that $s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right)$ , where $\varphi (n)$ is the number of integers positive cannot exceed $n$ and are relatively prime to $n$ . b) Prove that there is no integer $n \geq 2$ such that $s (n) = s (n + 2021)$