Equivalence Class for Abstract Algebra Class Let
$$R_3= \{(a,b)\mid a,b \in \mathbb{Z}\text{ and there exists }k \in \mathbb{Z} \text{ such that }a-b=3k\}.$$
I know there is an equivalence relation but I'm not 100% on what it means to be an equivalence class for this problem. In class we got 3: $\{0,3,6,9,\ldots\}$ and $\{1,4,7,10,-2,-5,\ldots\}$ and $\{2, 5, 8, 11, -1, -4,\ldots\}$. 
I don't understand where these cells came from. Help?

I'll try to put it this way:

Define a relation $\sim$ on $\mathbb Z$, such that $a \sim b \iff \exists k \in \mathbb Z ~~ \text{such that}~~~~a-b=3k$
What does this say?
Integers $a$ and $b$ are related if and only if on their difference is a multiple of $3$. Since, the remainder when $a-b$ is divided by $3$ is the difference of the remainders when $a$ and $b$ are divided by $3$, taken(all taken$\mod 3$).
So, integers $a$ and $b$ are related if and only if they leave the same remainder when divided by $3$.

Now try to put all those numbers that are related to each other in the same "cell" and those that are not related in different "cells".
But, now notice that the number of distinct cells you'll need for the purpose is no more than $3$ and no less! (Why?)
Construct these "cells" to see how they coincide with what you have written down in your class.
And, now call these cells "equivalence classes".
