Factorize RSA knowing several N and E I did a CRT challenge yesterday and there`s one problem I was unable to solve, probably due to my lack of understanding advanced crypto math.

It`s about RSA. There are ten given pairs of N and E (modulo and exponent). E is always the standard exponent 10001 (hex). N is a 2048 bit number.

The task: factorize all N.

Well, I tried everything known to me (CRT, Wiener, p-1), no luck. Now it occurred to me that maybe 10 pairs aren't given to keep you busy but that maybe the vulnerability is located there?

Any ideas?

no, I don't have the data anymore. :)
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.