RSA: Derive modulus from known ciphertexts encoded with same keypairs
Here is the hint you requested:


We know that we have $m_1^e - c_1 = k_1n$ for some integer $k_1$
We know that we have $m_2^e - c_2 = k_2n$ for some integer $k_2$


So, given that we know the values $k_1n$ and $k_2n$, how can we recover the common $n$ value?  Hint: look at the $\gcd$ function...