alternative proof of Euclid’s lemma
We give an alternative proof (see Euclid’s lemma proof), which does not use the Fundamental Theorem of Arithmetic (since, usually, Euclid’s lemma is used to prove FTA).
Lemma 1.
If and then .
Proof.
By assumption , thus we can use Bezout’s lemma to find integers such that . Hence and . Since and (by hypothesis), we conclude that as claimed.∎