Prime factorization of RSA modulus
If you knew $pq$ and $p+q$, you could find $p$ and $q$ by algebra. If you knew $pq$ and $(p-1)(q-1)$, you could find $p+q$. Now $e_1d_1-1$ and $e_2d_2-1$ are both said to be multiples of $\phi(n)$, so their greatest common divisor should be a (smaller) multiple of $\phi(n)$, from which you could easily guess $\phi(n)$.

I've probably said too much already, given that this is a homework assignment.