# m is said to be related to n if m and n are integers and m – n is divisible by 13. Does this define an equivalence relation?

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.

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