See Euclid's lemma on Wiktionary
Download JSON data for Euclid's lemma meaning in All languages combined (6.2kB)
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"etymology_text": "Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements.",
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"ref": "1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78",
"text": "I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that #x5C;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#x3D;2, the only even prime, and taking b#x3D;a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing #x5C;sqrt 2 to be irrational to prove that #x5C;sqrtp is irrational for any prime p.",
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"ref": "2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179",
"text": "If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: 12#x5C;mid 9#x5C;cdot 8, but 12#x5C;nmid 9 and 12#x5C;nmid 8.",
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"ref": "2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory, Taylor & Francis (Chapman & Hall / CRC Press), page 42",
"text": "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.",
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"The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;\nslightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;\n(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.",
"The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;"
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"slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;"
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"The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;\nslightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;\n(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.",
"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."
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"ref": "1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78",
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"ref": "2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179",
"text": "If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: 12#x5C;mid 9#x5C;cdot 8, but 12#x5C;nmid 9 and 12#x5C;nmid 8.",
"type": "quotation"
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"The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;"
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