Decrypting RSA encrypted messages with repeated use of p,q I require help with the following assignment. 

The Assignment 


  A company by the name "Secure Keys" is very good at creating random prime numbers, however a new CEO decides to cut costs and use random numbers more than once in generating RSA key pairs.
  
  100 messages from this company were intercepted, all messages contain the cyphertext and the modulo used for decryption, we also know that their RSA encryption uses e = 65537 (2^16 + 1)
  
  You are to decrypt the files and find the message containing the flag.


I'm lost in how to approach this problem, since I know that having the n,e isn't enough to decrypt a single message and I failed to connect the repetition on prime numbers used part.
The assignment, or at least the part of it that you've quoted, seems somewhat ambiguous, but I'm going to assume that it means that among the RSA moduli there are pairs $(n_A, n_B)$ that share a common prime factor $p$ (i.e. $n_A = p \cdot q_A$, $n_B = p \cdot q_B$, with $q_A \ne q_B$).

In that case, you can trivially find the shared prime factor by calculating the greatest common divisor of the two moduli $n_A$ and $n_B$, e.g. using the Euclidean algorithm.  Even if you don't know in advance which moduli share a common factor, you can simply test each pair of moduli: the GCD of any two numbers is always, by definition, a divisor of both of them, so if you find any two RSA moduli whose GCD isn't 1, you've just factored both of them.

Note that this is not just an artificial exercise, but a practical attack against poor quality RSA key generation implementations found in the wild.  For a demonstration of this attack (and more details on how to efficiently carry it out in practice), see e.g. the paper "Mining Your Ps and Qs: Detection of Widespread Weak Keys in Network Devices" by Heninger, Durumeric, Wustrow and Halderman (2012).

Ps. See also this related question from a few years ago.