Search results
Results from the WOW.Com Content Network
Appearance. In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is ...
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5) , and the same number 21 is also the GCD of 105 and 252 − 105 = 147 .
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that. This is a certifying algorithm, because the gcd is the only ...
Thus, the GCD is 2 2 × 3 = 12. The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, [1][2] is an algorithm that computes the greatest common divisor (GCD) of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with ...
17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 3, the quotient is 5, and the remainder is 2 (which is strictly smaller than the divisor 3), or more symbolically, 17 = (3 × 5) + 2. In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the ...
Lamé's Theorem is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm.Using Fibonacci numbers, he proved in 1844 [1] [2] that when looking for the greatest common divisor (GCD) of two integers a and b, the algorithm finishes in at most 5k steps, where k is the number of digits (decimal) of b.
As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as 3 = 15 × (−9) + 69 × 2, with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem , result from Bézout's identity.
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original ...