Solution:

At least two colors are needed in a good coloring of $P$. We show that two is sufficient. Write the positive integer $n$ as $p_{1}^{e_{1}} p_{2}^{e_{2}} \ldots p_{k}^{e_{k}}$, for distinct primes $p_{1}, p_{2}, \ldots, p_{k}$, and let $f(n) = e_{1} + e_{2} + \ldots + e_{k}$. Notice that if $v_{a}$ and $v_{b}$ are connected, then $f(a)$ and $f(b)$ have opposite parity. So, if we color $v_{n}$ red if $f(n)$ is odd and blue otherwise, the two-coloring is good.