Equivalence relation class $\bar{0}$

Question

In the set $\mathbb{Z}$ we define the following relation:

$$a\Re b \iff a\equiv \bmod2\text{ and }a\equiv \bmod3$$

1)Prove that $\Re$ is an equivalence relation. (Done)

2) Describe the equivalence class $\bar{0}$. How many different equivalence classes exist?

My thought on $\bar{0}$ is :

$$\bar{0} = a \in \mathbb{Z} / a\Re b \implies a\equiv 0\bmod2\text{ and }a\equiv 0\bmod3$$

Answer 1

Your thoughts?: exactly.

Now, $a \equiv 0 \pmod 2 $ and $a \equiv 0 \pmod 3 \implies a \equiv 0 \pmod 6$.

So, the equivalence class of $\bar{0}$ is equal to the set of all integer multiples of $6$: $$\bar{0} = \{6k\mid k\in \mathbb Z\}$$

Can you see that the equivalence classes of $\Re$ are the residue classes, modulo $6$?