Determining if the relation is an equivalence one.

Question

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?

Answer 1

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 $x\sim z$.

Answer 2

Hint note that $|x-z|=|x-y+y-z|$