Question 4
For an integer $n\ge2$, let $s(n)$ be the sum of the positive integers at most $n$ that are not relatively prime to $n$.
a) Prove that $s(n)=\frac n2\bigl(n+1-\varphi(n)\bigr)$, where $\varphi(n)$ counts the positive integers at most $n$ relatively prime to $n$.
b) Prove that no integer $n\ge2$ satisfies $s(n)=s(n+2021)$.
a) Prove that $s(n)=\frac n2\bigl(n+1-\varphi(n)\bigr)$, where $\varphi(n)$ counts the positive integers at most $n$ relatively prime to $n$.
b) Prove that no integer $n\ge2$ satisfies $s(n)=s(n+2021)$.