multiples of subset of $\mathbb{N}$

Question

Suppose that $ A_1\cup \dotsm A_n=\mathbb{N}$ is a partition of $\mathbb{N}$ into disjoint subsets. Is it true that there is an integer $1 \leq k \leq n$ such that the set $A_k\cap 2A_k$ is infinite?

Answer 1

Write each $N=2^{qn+r}m$ where $m$ is odd and $0\leq r<n$. Then put $N$ in $A_{r+1}$. Then $A_k\cap 2A_k$ is empty for all $k$.