BG-MO 2004 Problem 2

For any positive integer $n$ the sum $1+\frac 12+ \cdots + \frac 1n$ is written in the form $\frac{P(n)}{Q(n)}$ , where $P(n)$ and $Q(n)$ are relatively prime.

a) Prove that $P(67)$ is not divisible by 3;

b) Find all possible $n$ , for which $P(n)$ is divisible by 3.