I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that √2 is irrational. I said there that if 2 is a factor of a² then a itself must be even. This follows from Euclid's Lemma upon taking p=2, the only even prime, and taking b=a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing √2 to be irrational to prove that √p is irrational for any prime p.