Limit of expected number of squarefree divisiors of gcd as n approaches infinity

Question

Let $X_n$ and $Y_n$ be two random numbers, drawn uniformly and independently from {1,…,n}. Find $\lim _ {n \rightarrow \infty}$ of the expected value of the number of squarefree divisors of gcd($X_n, Y_n)$.

I found that the squarefree conditions means that the number of squarefree divisiors is $a ^k$ where $a$ is the amount of prime divisors. I'm a bit stuck though now.