R = {(a, b) : a – b is divisible by n} on Z.

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Let a ∈ Z

⇒ a – a = 0 × n

⇒ a – a is divisible by n

⇒ (a, a) ∈ R

⇒ R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let (a, b) ∈ R

⇒ a – b = np For some p ∈ Z

⇒ b – a = n(–p)

⇒ b – a is divisible by n

⇒ (b, a) ∈ R

⇒ R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Let (a, b) ∈ R and (b, c) ∈ R

⇒ a – b = np and b – c = nq For some p, q ∈ Z

⇒ a – c = n (p + q)

⇒ a – c = is divisible by n

⇒ (a, c) ∈ R

⇒ R is transitive

∴ R being reflexive, symmetric and transitive on Z.

⇒ R is an equivalence relation on Z