See Euclid's lemma on Wiktionary
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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 #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.",
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"text": "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.",
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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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Download raw JSONL data for Euclid's lemma meaning in All languages combined (4.7kB)
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