Question 63
For each integer $n\ge2$, let $s(n)$ denote the sum of all positive integers at most $n$ that are not relatively prime to $n$.
Part (a). Prove that
$$
s(n)=\frac n2\bigl(n+1-\varphi(n)\bigr),
$$
where $\varphi(n)$ is the number of positive integers at most $n$ that are relatively prime to $n$.
Part (b). Prove that there is no integer $n\ge2$ such that
$$
s(n)=s(n+2021).
$$
Part (a). Prove that
$$
s(n)=\frac n2\bigl(n+1-\varphi(n)\bigr),
$$
where $\varphi(n)$ is the number of positive integers at most $n$ that are relatively prime to $n$.
Part (b). Prove that there is no integer $n\ge2$ such that
$$
s(n)=s(n+2021).
$$