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

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Let m ∈ Z

⇒ m – m = 0

⇒ m – m is divisible by 13

⇒ (m, m) ∈ R

⇒ R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let m, n ∈ Z and (m, n) ∈ R

⇒ m – n = 13p For some p ∈ Z

⇒ n – m = 13 × (–p)

⇒ n – m is divisible by 13

⇒ (n – m) ∈ 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 (m, n) ∈ R and (n, q) ∈ R For some m, n, q ∈ Z

⇒ m – n = 13p and n – q = 13s For some p, s ∈ Z

⇒ m – q = 13 (p + s)

⇒ m – q is divisible by 13

⇒ (m, q) ∈ R

⇒ R is transitive

Hence, R is an equivalence relation on Z.