\indent 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.