MN-MO 2007 Problem 2

For all $n\ge2$ , let $a_n$ be the product of all coprime natural numbers less than $n$ . Prove that

(a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$ (b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$ Here $p$ is an odd prime number and $\alpha\in\mathbb N$ .