HMMT 2013 T9

Let $m$ be an odd positive integer greater than 1 . Let $S_{m}$ be the set of all non-negative integers less than $m$ which are of the form $x+y$, where $x y-1$ is divisible by $m$. Let $f(m)$ be the number of elements of $S_{m}$. (a) Prove that $f(m n)=f(m) f(n)$ if $m, n$ are relatively prime odd integers greater than 1 . (b) Find a closed form for $f\left(p^{k}\right)$, where $k>0$ is an integer and $p$ is an odd prime.