Describing A Congruence Class The question is, "Give a description of each of the congruence classes modulo 6."
Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The relation would then be $R=\{(x,y)|x \equiv y~(mod~6)\}$
Then, $[n]_6 =\{x \in Z|~x \equiv n~(mod~6)\}$
$[n]_6=\{x \in Z|~6|(x-n)\}$
$[n]_6=\{x \in Z|~k(x-n)=6\}$, where $n \in Z$
As I looked over what I did, I started think that this would not describe all of the congruence classes on modulo 6.  Also, what would I say k is? After despairing, I looked at the answer key, and they talked about there only being 6 equivalence classes. Why are there only six of them? It also says that you can describe equivalence classes as one set, how would I do that?

You're right, it does not describe all the congruence classes -- due to an error. Namely, notice $\rm\:6\mid x\!-\!n\:$ means $\rm\:6\color{#C00}k = x\!-\!n,\:$ not $\rm\:6 = \color{#C00}k(x\!-\!n).\:$ Recall $\rm\:a\mid b \!\iff\! a\color{#C00}k = b\:$ for some $\rm\:\color{#C00}k\in\Bbb Z.$ Thus $\rm\:x\in [n]_6$ $\!\iff\!$ $\rm\exists k\in \Bbb Z\!:\ x = n + 6k$ $\!\iff\!$ $\rm x\in n + 6\,\Bbb Z = \{\ldots n\!-\!12,\ n\!-\!6,\ n,\ n\!+\!6,\ n\!+\!12,\ldots\}.$
