discrete mathematics - Prove if a (mod n) has a multiplicative inverse, then it's unique

Assume that an integer $a$ has a multiplicative inverse modulo an integer $n$. Prove that this inverse is unique modulo $n$.

I was given a hint that proving this Lemma: $$\begin{align} n \mid ab \ \wedge \ \operatorname{gcd}\left(n,a\right) = 1 \qquad \Longrightarrow \qquad n \mid b \end{align}$$ should help me in finding the answer.

Here are my steps in trying to solve the problem: $$\begin{align} \operatorname{gcd}\left(n,a\right) = 1 & \qquad \Longrightarrow \qquad sn + ta = 1 \qquad \Longrightarrow \qquad sn = 1 - ta \\ & \qquad \Longrightarrow \qquad 1 \equiv ta \mod n \qquad \Longrightarrow \qquad ta \equiv 1 \mod n . \end{align}$$ I know that having the GCD of m and a equal to 1 proves there is a multiplicative inverse mod n, but I'm not sure on how to prove $n \mid b$ with and how it helps prove the multiplicative inverse is unique.

Answer

For the first part note as $(n,a) = 1$, then there exist $x,y \in \mathbb{Z}$, s.t. $nx + ay = 1$. Multiply both sides by $b$ and you will get $nxb + (ab)y = b$. The LHS is obviously divisible by $n$, so then the RHS must be too. Hence $n \mid b$.

This thing proves that the inverse exist. To note that it's unique modulo $n$ assume that $aa_1 \equiv 1 \pmod n$ and $aa_2 \equiv 1 \pmod n$. Then we have that there exist integer $x,y$ s.t. $aa_1 + nx = 1$ and $aa_2 + ny = 1$. Subtracting the two equations we have that $a(a_1 - a_2) = n(y-x) \implies a(a_1 - a_2) \equiv 0 \pmod n \implies a_1 \equiv a_2 \pmod n$. Hence the multiplicative inverse is unique in $\mathbb{Z}_n$