1. Equivalence Class Representatives

Okay, so this question seemed easy to me earlier, but now i'm getting paranoid about it.

Problem:

Define a relation $\displaystyle \sim$ on the set $\displaystyle \mathbb{N}$ of natural numbers by

$\displaystyle a \sim b$ iff $\displaystyle a = b \cdot 10^k$ for some $\displaystyle k \in \mathbb{Z}$

(b) Give a complete set of equivalence class representatives

Things That Might Come In Handy:

I suppose whoever helps me with this will know what an equivalence class is, so i will tell you about equivalence class representatives.

let $\displaystyle S$ be a set which we have defined an equivalence relation on. The set of equivalence class representatives is the subset of $\displaystyle S$ containing precisely one element from each equivalence class.

so, for example. lets say i have the set $\displaystyle \mathbb{Z}$, and my relation on this set is

for $\displaystyle a,b \in \mathbb{Z}$, $\displaystyle a \sim b$ iff $\displaystyle a$ and $\displaystyle b$ have the same parity.

clearly this partitions the integers into the set of even integers and the set of odd integers. so if $\displaystyle e$ is any even integer, its equivalence class is $\displaystyle [e] = \{ 0,~ \pm 2,~ \pm 4, \cdots \}$ and if $\displaystyle o$ is any odd integer, its equivalence class is given by $\displaystyle [o] = \{ \pm 1,~ \pm 3,~ \pm 5, \cdots \}$

for the set of equivalence class representatives, we would simply choose any one even number and any one odd number, so $\displaystyle \{ 0,1 \}$ suffices.

What I have tried:

I realize for any $\displaystyle a \in \mathbb{N}$, $\displaystyle [a] = \{a \cdot 10^k | k \in \mathbb{N} \}$

but it gets more complicated. if $\displaystyle a$ happens to be a multiple of 10, then for some of the k's in the above set, -k also works, but not all. secondly, what would i choose for the set of representatives. i was thinking about choosing $\displaystyle \mathbb{N}$, but this will cause repeats, as for example, 1, 10, 100, 1000 etc will be in there when all are in [1] and [10] and [100] etc. how would i define my set here?

2. i think $\displaystyle \{ n ~|~ n \in \mathbb{N} \mbox{ and } n \mbox{ is not a multiple of } 10 \}$ works.

does it?