Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.
Problem
Source: III International Festival of Young Mathematicians Sozopol 2012, Theme for 10-12 grade
Tags: number theory, Sets