Let R be a relation on the set of integers given by $a\, R\,b⇒a = 2^k b$ for some integer k. Then, R is

an equivalence relation

For any integer a, we have

$a=2^0a$

$⇒ a = 2^ka$, where k = 0

$⇒ (a, a) ∈ R$

So, R is reflexive on Z.

Let $(a, b) ∈ R$. Then,

$a = 2^k b$ for some integer k

$⇒ b=2^{-k}$ a for some integer k

$⇒ (b, a) ∈ R$.

So, R is symmetric on Z.

Let $(a, b) ∈ R$ and $(b, c) ∈ R$.

Then, $a = 2^k b$ and $b = 2^m c$ for some integers k and m

$⇒a=2^{k+m}c$

$⇒(a, c) ∈ R$

So, R is transitive on Z.

Hence, R is an equivalence relation on Z.