Factorize RSA knowing several N and E
If the moduli share no commonness / weak properties, it's impossible to solve the problem as given at hand. (As CodesInChaos pointed out)

Now as you're given RSA moduli, which are always constructed as $n=pq$, you can be sure that if you find $gcd(n_1,n_2)=R&gt;1$, than either $R=p$ or $R=q$ must hold.
As you pointed out in the comments you indeed found such an $R_1$ for $n_1,n_2$. Now you reconstruct $P_1=n/R_1$ and check if $P_1*R_1=n$ holds, which is the case.
In the next step you check wether $P_1$ or $R_1$ divide any of the other moduli.
If it does, you reconstruct the other prime of these moduli via $n/R_1$. You repeat this procedure for all factors you're given.
As soon as you've done this step and still have unfactored moduli left, you try to apply the GCD on all pairs of unfactored moduli and use the resulting factors to obtain the remaing factors and factor the remaining moduli.