  We examine Euclid's lemma that if $p$ is a prime number such that $p | ab$,
then $p$ divides at least one of $a$ or $b$. Specifically, we consider the
common misapplication of this lemma to numbers that are not prime, as is often
made by undergraduate students. We show that a randomly chosen implication of
the form $r |ab \Rightarrow r|a \text{ or } r|b$ is almost surely false in a
probabilistic sense, and we quantify this with a corresponding asymptotic
formula.
