Prove that 4|(x-y) is relation of equivalency?

Question

Prove that 4|(x-y) is relation of equivalency.

This is gonna be on my exam in 3 hours T_T how do I answer it?

Answer 1

Reflexive: for any x, 4|(0 - 0) since 4.0 = 0.

Symmetric: for any x, y, 4|(x-y)
<=> for some integer k, (x-y) = 4k
<=> for some integer (-k), (y-x) = 4(-k)
<=> 4|(y-x)

Transitive: Suppose 4|(x-y) and 4|(y-z). Then there exist k and l such that x-y = 4k and y-z = 4l. Then x-z = (x-y) + (y-z) = 4(k+l), so 4|(x-z).

So this relation is reflexive, symmetric and transitive, so it is an equivalence relation.