Question 75
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 $k\gt 0$ is an integer and $p$ is an odd prime.
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 $k\gt 0$ is an integer and $p$ is an odd prime.