Is it possible to recover an RSA modulus from its signatures? Let's say that you have some small number of RSA signatures of known data: you know some pairs $(m_k, c_k)$ such that ${c_k}^e \equiv m_k \pmod n$.  If you know $e$, because probably it's one of $\{3, 17, 65537\}$, is it possible to recover $n$?
calculate
$$gcd(c_1^e - m_1 , c_2^e- m_2, \dots , c_k^e-m_k)$$

With a bit luck this should get $n$.
It will be an interesting exercise to calculate the probability of success based on the number of cleartect/ciphertext pairs.