On the Success of Mishandling Euclid's Lemma

We examine Euclid’s lemma that if $p$ is a prime number such that $p\mathrm{\mid }ab$, 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 $r\mathrm{\mid }ab\Rightarrow r\mathrm{\mid }a\text{ or }r\mathrm{\mid }b$ is almost surely false in a probabilistic sense, and we quantify this with a corresponding asymptotic formula.