How to compute $\mathbb{Z}_n^*$, the unit group of the integers modulo $n$? I am trying to wrap my head around calculating residue classes, for example $\mathbb{Z}^*_{12} = [1]...$
Would anyone mind making up an example with a decent explanation. I just wrap my head around things easier seeing the process.
Thanks

Since you clarified in a comment below your post, you seem interested in knowing how to determine which elements are in the unit group of $\mathbb Z_{12}$, and perhaps, in general how to compute the elements of any unit group $\mathbb Z_n$:
The unit group of $\mathbb Z_{12}$, denoted $\mathbb Z_{12}^*$, consists of all residue classes of elements $z \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ such that $z$ is relatively prime to $12$, all $z$ such that $\gcd(z, 12) = 1$.
Since the only elements $z \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ that are relatively prime to $12$ are exactly $1, 5, 7, 11$, we have that $$\mathbb Z_{12}^* = \{[1], [5], [7], [11]\}$$ which is usually written simply as $$\mathbb Z_{12}^* = \{1, 5, 7, 11\}$$
In general, for a unit group $\mathbb Z_n^*$, Let $[n-1] = \{1, 2, \cdots, n-1\}$,
$$\mathbb Z_n^* = \{z \in [n-1] \mid \gcd(z, n) = 1\}$$
