Question 30
Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n \lt 60$ (both Kelly and Jason know that $n \lt 60$, but they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason the number of divisors of $n$. Then, Kelly and Jason have the following dialogue:
Kelly: I don't know what $n$ is, and I'm sure that you don't know either. However, I know that $n$ is divisible by at least two different primes.
Jason: Oh, then I know what the value of $n$ is.
Kelly: Now I also know what $n$ is.
Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $n$?
Kelly: I don't know what $n$ is, and I'm sure that you don't know either. However, I know that $n$ is divisible by at least two different primes.
Jason: Oh, then I know what the value of $n$ is.
Kelly: Now I also know what $n$ is.
Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $n$?