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?

Let’s start with your correct description
$$[n]_6=\{x\in\Bbb Z:x\equiv n\!\!\!\pmod 6\}=\{x\in\Bbb Z:6\mid x-n\}$$
and actually calculate $[n]_6$ for some values of $n$. 


*

*$[0]_6=\{x\in\Bbb Z:6\mid x-0\}=\{x\in\Bbb Z:6\mid x\}=\{x\in\Bbb Z:x=6k\text{ for some }k\in\Bbb Z\}$; this is just the set of all multiples of $6$, so $[0]_6=\{\dots,-12,-6,0,6,12,\dots\}$.

*$[1]_6=\{x\in\Bbb Z:6\mid x-1\}=\{x\in\Bbb Z:x-1=6k\text{ for some }k\in\Bbb Z\}$; this isn’t quite so nice, but we can rewrite it as $\{x\in\Bbb Z:x=6k+1\text{ for some }k\in\Bbb Z\}$, the set of integers that are one more than a multiple of $6$; these can be described as the integers that leave a remainder of $1$ when divided by $6$, and $[1]_6=\{\dots,-11,-5,1,7,13,\dots\}$. 
More generally, if $x$ is any integer, we can write it as $x=6k+r$ for integers $k$ and $r$ such that $0\le r<6$: $r$ is the remainder when $x$ is divided by $6$. Then 
$$\begin{align*}
[r]_6&=\{x\in\Bbb Z:6\mid x-r\}\\
&=\{x\in\Bbb Z:x-r=6k\text{ for some }k\in\Bbb Z\}\\
&=\{x\in\Bbb Z:x=6k+r\text{ for some }k\in\Bbb Z\}\\
&=\{6k+r:k\in\Bbb Z\}\;
\end{align*}$$
the set of all integers leaving a remainder of $r$ when divided by $6$. You know that the only possible remainders are $0,1,2,3,4,5$, so you know that this relation splits $\Bbb Z$ into exactly six equivalence classes, $[0]_6,[1]_6,[2]_6,[3]_6,[4]_6$, and $[5]_6$.
