On the Success of Seriously Mishandling Euclid's Lemma

摘要

In the November 2016 issue of the Monthly, A. W. Dudek showed that the implication r vertical bar ab -> r vertical bar a or r vertical bar b over positive integers is almost always false in a certain probabilistic sense. By considering two variations on this idea, we show that this fact continues to hold when ab is replaced by a product with a fixed number of factors, but a nontrivial frequency develops when r is assumed to have a fixed number of prime divisors, including multiplicity.