Find quotient space on $\mathbb{N} $ On $\mathbb{N}$ is given equivalence relation R with $nRm \iff 4|n-m$. Topology on $\mathbb{N}$ is defined with $\tau=\{\emptyset\}\cup\{U\subseteq\mathbb{N}|n\in U \wedge m|n \implies m\in U\}$.
I need to find quotient space $(\mathbb{N}/R,\tau_{R})$.
I have solution: $\tau_{R}=\{\emptyset,\mathbb{N}/R,\{[1],[2],[3]\}\}$ where $\mathbb{N}/R=\{[1],[2],[3],[4]\}$.
But I have no idea how to prove that $p^{-1}[\{[1],[2],[3]\}]=\cup_{k\in \mathbb{N}_0}\{4k+1,4k+2,4k+3\}$, where $p$ is quotient mapping, contains all divisors of its elements. 
(for other set it's easy to find element whose divisor is not in set)

Note that $p^{-1}(\{[1],[2],[3]\}) = \mathbb{N}\backslash 4\mathbb{N}$.
We need to prove that $n\in\mathbb{N}\backslash 4\mathbb{N}$ and $m|n$ implies $m\in\mathbb{N}\backslash 4\mathbb{N}$ and we do this by contraposition.
Suppose $m\notin\mathbb{N}\backslash 4\mathbb{N}$, then $m = 4k$ and thus $m|n$ implies $n = pm = 4pk$, hence $n\notin\mathbb{N}\backslash 4\mathbb{N}$ concluding the argument.
