Central to many factoring algorithms in use today is the following random process: generate random numbers in the interval [1,N] until some subset has a product which is a square. Naive probabilistic models for the distribution of prime factors suggest that this stopping time has a sharp threshold. Based on more sophisticated probabilistic models, we find a rigorous upper bound that is within a factor of 4/pi of a proven lower bound, and conjecture that our upper bound is in fact asymptotically sharp. This is joint work with Andrew Granville, Ernie Croot and Prasad Tetali.