CN-SMO 2013 Problem 6

$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$ . Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$ . Find all positive integers $x$ , such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$ divisors