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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 greatest common divisor is not unique: if d is a GCD of p and q, then the polynomial f is another GCD if and only if there is an invertible element u of F such that = and =. In other words, the GCD is unique up to the multiplication by an invertible constant.
Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. 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.
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm (a, b), is the smallest positive integer that is divisible by both a and b. [1][2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both ...
The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c. [6] The greatest common divisor can be visualized as follows. [7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly.
Lamé's theorem. 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 5 k steps, where k is the number of digits (decimal) of ...
Möbius (or Moebius) function mu (n). mu (1) = 1; mu (n) = (-1)^k if n is the product of k different primes; otherwise mu (n) = 0. The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. [i][ii][2] It is ubiquitous in elementary and ...
where both A n (z) and B n (z) have integer coefficients, A n (z) has degree φ(n)/2, and B n (z) has degree φ(n)/2 − 2. Furthermore, A n (z) is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, B n (z) is palindromic unless n is composite and ≡ 3 (mod 4), in which case it is antipalindromic.