Multiplicative group of integers modulo n definition issues 
It is easy to verify that the set $(\mathbb{Z}/n\mathbb{Z})^\times$ is closed under multiplication in the sense that $a, b ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $ab ∈ (\mathbb{Z}/n\mathbb{Z})^\times$, and is closed under inverses in the sense that $a ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $a^{-1} ∈ (\mathbb{Z}/n\mathbb{Z})^\times$.

The question is the following:
Firstly, are $a$ and $b$ referring to each equivalence class of integers modulo $n$?
Secondly, by $a^{-1}$, what is this referring to? If $a$ is the equivalence class, I cannot see (or I am not sure) how I can make inverse set.

It is also standard to identify $(\mathbf Z/n \mathbf Z)$, which is typically defined algebraically as a set of equivalence classes, with the integers $\{0, \dots, n-1 \}$. One thing to note is that $(\mathbf Z/n \mathbf Z)$ is essentially finitary, so the representation as integers is sometimes useful (but less convenient algebraically)
