Search results
Results from the WOW.Com Content Network
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 ...
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 ...
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 a = bq + r. and 0 ≤ r < |b|, where |b| denotes the absolute value of b. [4]
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.
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
A Wisconsin woman is taking legal action after learning a feeding tube was allegedly left inside her body during surgery nearly 35 years ago. In 1989, Deborah Lowe was pregnant with twins when she ...
Starting from two polynomials a and b, Euclid's algorithm consists of recursively replacing the pair (a, b) by (b, rem(a, b)) (where "rem(a, b)" denotes the remainder of the Euclidean division, computed by the algorithm of the preceding section), until b = 0. The GCD is the last non zero remainder.