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Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. For example, if p = 19 , a = 133 , b = 143 , then ab = 133 × 143 = 19019 , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well.
Filip Saidak gave the following proof by construction, which does not use reductio ad absurdum [15] or Euclid's lemma (that if a prime p divides ab then it must divide a or b). Since each natural number greater than 1 has at least one prime factor , and two successive numbers n and ( n + 1) have no factor in common, the product n ( n + 1) has ...
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. [13] The proof that follows is inspired by Euclid's original version of the Euclidean algorithm. Assume that is the smallest positive integer which is the product of prime numbers in two different ways.
Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. [118] For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common ...
Any theorem in Euclid's Elements, and in particular: Euclid's theorem that there are infinitely many prime numbers; Euclid's lemma, also called Euclid's first theorem, on the prime factors of products; The Euclid–Euler theorem characterizing the even perfect numbers; Geometric mean theorem about right triangle altitude
Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses Euclid's lemma, which requires factors to be prime rather than just irreducible. Indeed, one has the following characterization: let A be an integral domain. Then the following are equivalent. A is a UFD.
Euclidean division is based on the following result, which is sometimes called Euclid's division lemma. Given two integers a and b , with b ≠ 0 , there exist unique integers q and r such that
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".