Let $n$ be a positive integer and $a,b$ be invertible integers modulo $n$ such that $a\equiv b^{-1}\pmod n$. What is the remainder when $ab$ is divided by $n$?
I know that $a\equiv b^{-1}\pmod n$ means that $ab\equiv 1\pmod n$, so the remainder when $ab$ is divided by $n$ must be $1$. To check, I can multiply both sides of $a\equiv b^{-1}\pmod n$ by $b$ and get $ab\equiv b^{-1}b\equiv 1\pmod n$. 