If $n$ is a positive number, let $\mathbb{Z}_n^*$ be the subset of class representatives in $\{1,..., n-1\}$ which are relatively prime to $n$. Show that if $[a],[b] \in \mathbb{Z}_n^* $ then $[ab] \in \mathbb{Z}_n^*$.
I think $[a]$ is an equivalence class. But what does it represent? If $a = 1$, what does $[1]$ imply?
Update: this is what I understand so far. For example, let n = 5, a = 2. Then relatively primes of n is {1,2,3,4}, and [a] = {2,4} because it's only 2 in {1,2,3,4} that has residual when divided by n = 5. Is this correct?
Update: My thought on proving the problem. Let [a] = {a + kn | k $\in \mathbb{Z}_n^*$}, [b] = {b + ln | l $\in \mathbb{Z}_n^*$}. Pick an a' from [a] and b' from [b]. I can say a' = a + kn, b' = b +ln. Then a'*b' = ab + n(ak + bk + $nk^2$). This is congruent to ab mod n. Therefore [ab] $\in \mathbb{Z}_n^*$. Would this be an appropriate proof?
In $\mathbb{Z}$, we define an equivalence relation by : $$ a \text{ and } b \text{ are equivalent } \iff n \mid b-a $$ so $[a]=\{ b\in \mathbb{Z} : a \text{ and } b \text{ are equivalent }\}=\{ b \in \mathbb{Z} : n|b-a\}=\{ b\in \mathbb{Z} : b-a=nk \text{ s.t } k\in \mathbb{Z}\}=\{a+nk : k\in \mathbb{Z}\},$
Thus in $\mathbb{Z}_3$, for example $[1]=\{1+3k : k\in \mathbb{Z}\}=\{\cdots,-2,1,4,7,\cdots \}.$