Let $m$ and $n$ be non-negative integers. If $m = 6n + 2$, then what integer between $0$ and $m$ is the inverse of $5$ modulo $m$? Answer in terms of $n$.

Recall that the inverse of $a$ modulo $m$ is an integer $b$ such that $ab \equiv 1 \pmod{m}$. In this case, we are looking for the integer $b$ such that $5b \equiv 1 \pmod{6n+2}$.

Since we are given that $m = 6n + 2$, we can rewrite the congruence as $5b \equiv 1 \pmod{6n + 2}$. Multiplying both sides by $5^{-1}$ (the inverse of $5$ modulo $6n + 2$) gives $b \equiv 5^{-1} \pmod{6n + 2}$.

Therefore, the integer between $0$ and $m$ that is the inverse of $5$ modulo $m$ is $5^{-1}$, which is $5^{-1} \equiv \boxed{5n + 1} \pmod{6n + 2}$.