Exploit RSA , given some information about n

Question

Every week Bob updates his public RSA modulus $n=p*q$ and encryption exponent $e$ by picking a new pair of primes $p$ and $q$ and a new exponent $e$. After several weeks, bob becomes lazy and decides to keep the prime $p$ fixed and only updates the other prime $q$ every week. Eve hears about this. How can she exploit this information to decrypt messages sent to Bob using his public key ?

I don't know how to solve it, any help or hint would be much appreciated.
I understand we need to retrieve $p$ and then we can get any $q_i$, and from there decipher the messages, but how can we do that ? okay so $n_1 = pq_1, n_2 = pq_2 , n_3 =pq_3$ , so on ... How do we get $p$

Answer 1

The question spezifies $n_1 = p*q_1$, $n_2 = p*q_2$. With that we can calculate $gcd(n_1,n_2) = p$ which lets us solve $n_1 / p = q_1$ and $n_2 / p = q_1$. Now we have every information needed to calculate the private part of Bob's key and thereby decrypt all messages sent to him.