Determining if the relation is an equivalence one. Determine if the relation : $$x \sim y \iff |y-x| \text{ is  an  integer  multiple  of } 3$$
is an equivalence one.
Now, I think this is an equivalence relation but I am having troubles formally proving the transitivity.
Any help?

Suppose that $x\sim y$ and $y\sim z$. 
Then $3$ divides $|x-y|$. It follows that $3$ divides $x-y$.
Similarly, $3$ divides $y-z$.
So $3$ divides $(x-y)+(y-z)$. It follows that $3$ divides $x-z$, and therefore $3$ divides $|x-z|$. We conclude that $x\sim z$.
