Let R be a relation on $\displaystyle \mathbb{N}$, where $\displaystyle xRy \Leftrightarrow $$\displaystyle x\equiv y$ mod 8

I have an answer saying this is an equivalence relationship; however, I can't seem to derive it myself. Maybe it is because my understanding of what mod 8 means... the way I understand the mod 8 term is that x = y with a remainder of 8 (or multiple of 8?), but why has $\displaystyle \equiv$ been used instead of $\displaystyle =$? Does it make a difference?

To check if the relation is reflexive would I say x = x + 8k for some integer k? It seems like that would not be true or only for k = 0. I'm am stuck for symmetry as well.

I think I can see transitive... If

$\displaystyle x = y + 8k $

and

$\displaystyle y = z + 8m $

then

$\displaystyle x = (z + 8m) + 8k$

$\displaystyle = z + 8(m + k)$

So there is a remainder that is a multiple of 8.