How to show or prove equivalence relation?

Question

I have this relation :
for all integers m and n so :
m R n ⇔ m ≡ n mod(3)

How can I show that R is an equivalence relation

Answer 1

Here are some hints:

General: First note that $a\equiv b\pmod{3}\iff a-b\equiv0\pmod{3}\iff3|(a-b)$.

Reflexive: Everything divides zero so $3|(a-a)$ which implies that...

Symmetric: If $aRb$ then $3|(a-b)$. What can you say about $b-a$?

Transitive: If $aRb$ and $bRc$ then $3|(a-b)$ and $3|(c-b)$. Can you find a proof that this necessarily implies that $3|(c-a)$?