Prove that: If $n \in \mathbb{Z}_{m} \setminus \left\{0\right\}$ has a multiplicative inverse, then this is definitely unique

Question

Prove that: If $n \in \mathbb{Z}_{m} \setminus \left\{0\right\}$ for $m \in \mathbb{N}$ has a multiplicative inverse, then this is definitely unique.

I don't know where we can start with this proof but I think it's important to know that $n \in \mathbb{Z}_{m}$ has got a multiplicative inverse in $\mathbb{Z}_{m}$ if and only if the $\text{gcd }(n,m)=1$. That's what I have learned some days ago here, from many answers.

Maybe it's somehow possible to deduce it from this fact in order to prove it? Because that's what comes to my mind when we talk about multiplicative inverses :P

Answer 1

Hint If $k,l$ are multiplicative inverses of $n$ then what is $$knl=?$$