Euclid's lemma

alternative Euclid's Lemma

1. uncountablemathematicsThe proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;

   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.

   2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179

   2 more 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.

   1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78

   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.

   2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory, Taylor & Francis (Chapman & Hall / CRC Press), page 42

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.
   1. uncountablemathematicsslightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
   2. uncountablealgebrathe 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.