Analysis. The proof just given is conceptually even simpler than the original proof due to Euclid, since it does not use Eudoxus's method of "reductio ad absurdum," proof by contradiction. And unlike most other proofs of the theorem, it does not require Proposition 30 of Elements (sometimes called "Euclid's Lemma") that states: if p is a prime and p\ab, then either p\a or p\b. Moreover, our proof is constructive, and it gives integers with an arbitrary number of prime factors.