Meaning of Euclid's lemma | Babel Free

Definitions

1. The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
   uncountable
2. The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both; slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c; (algebra, by generalisation) the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.
   uncountable
3. slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
   uncountable
4. the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.
   uncountable

Examples

“I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that #92;sqrt 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#61;2, the only even prime, and taking b#61;a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing #92;sqrt 2 to be irrational to prove⟳ that #92;sqrtp is irrational for any prime p.”

“If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail⟳ to hold⟳. A specific example: 12#92;mid 9#92;cdot 8, but 12#92;nmid 9 and 12#92;nmid 8.”

“In our discussion of Euclid's lemma (Corollary 2.18), we noted that the uniqueness of factorization of integers is a fact that we often take⟳ for granted given the way it is introduced in school.”

CEFR level