$aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$ [Prove Equivalence Relation]

Question

The question:

$R$ is a relation on $\mathbb{N}$ defined by $aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$.

Prove that $R$ is an Equivalence relation.

The problem:

I can define an equivalence relation (Sym,Ref,Trans) and can prove basic problems of this type, like "Define a relation on $\mathbb{Z}$ by $xRy$ iff $x + 2y$ is divisible by $3$."

I'm getting stuck on how to incorporate the $10^k$.

This is all I have for reflexivity: $x=x*10^k$ is $xRx$ when $k = 0$.

Please help.

Answer 1

If $a=b(10)^k$ then $b=a(10)^{-k}$ is symmetry, if $a=b(10)^k$ and $b=c(10)^l$ then $a=c(10)^{k+l}$ is transitivity.