Cracking

Key Attacks
Try all possible public keys with modulus n.  Decrypt the message using each key, and hope you recognize the plain text  when you see it.

Message Attacks
Try and encrypt all possible messages that might be sent using the known public key.  If any yield the cypher text, you win.

How
Factor the known modules m into p and q.   From p,q, and e generate a d.  The private key is (m, d).

How hard is factoring?
The best general-purpose factoring algorithm today is the Number Field Sieve, which runs in time approximately
                                  O(e^(1.9223(ln n ^1/3)(ln ln n^(2/3)))).
 

Digits in Prime	Relative Work
3	3
30	73
60	174
90	284

Is factoring known to be hard?
Nope, and that's not expected.

Is RSA vulnerable without factoring?
It's not known to be, but various implementations have specific flaws.