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Recursively define a sequence of $B$'s as follows. Initially, each is empty. At each step $n > 1$, place $n$ in the first $B$ that contains only elements coprime to $n$. Clearly, for each prime $p$, there is no $B$ that contains two distinct multiples of $p$. Now fix a prime $p$ and a natural number $N > 1$, and consider the first $B$ that contains no multiple of $p$ after step $N$ has completed. The first power $p^k$ of $p$ that is larger than $N$ cannot be placed in any earlier $B$ (since all have a multiple of $p$), so it will be placed in $B$ if no multiple $p d$ of $p$ with $N < p d < p^k$ has been.