Word.ws English Dictionary

Euclid's lemma

Euclid's lemma noun: the proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;.

Noun

Euclid's lemma (uncountable)

  1. The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
    • 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....1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78
    • 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. slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
  3. 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.

Examples of Euclid's lemma in a sentence

  • 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.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:

Forms of Euclid's lemma

alternative formEuclid's Lemma

Origin of Euclid's lemma

Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements.

Questions about Euclid's lemma

What does Euclid's lemma mean?
Euclid's lemma means the proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;.
How do you use Euclid's lemma in a sentence?
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.
What part of speech is Euclid's lemma?
Euclid's lemma is a noun.

Words near Euclid's lemma in the dictionary

  1. euclase
  2. euclases
  3. Euclataw
  4. Euclid
  5. Euclid's algorithm
  6. Euclid's lemma
  7. euclidean
  8. Euclidean algorithm
  9. Euclidean algorithms
  10. Euclidean alphabet
  11. Euclidean distance

Source and license

Adapted from Wiktionary: Euclid's lemma (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

Last updated:

Cite this entry: “Euclid's lemma.” Word.ws, https://word.ws/euclid%27s_lemma. Based on Wiktionary, CC BY-SA 4.0.