HMMT 2009 T2

(a) [6] Let $P$ be a graph with one vertex $v_{n}$ for each positive integer $n$. If $a<b$, then an edge connects vertices $v_{a}$ and $v_{b}$ if and only if $\frac{b}{a}$ is a prime number. What is the chromatic number of $P$ ? Prove your answer.