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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 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.
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 arithmetic shifts, comparisons ...
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.
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 ...
Say we want to obtain the GCD of the two integers a and b. Let a ≥ b. If b contains only one digit (in the chosen base, say β = 1000 or β = 2 32), use some other method, such as the Euclidean algorithm, to obtain the result. If a and b differ in the length of digits, perform a division so that a and b are equal in length, with length equal ...
Construct an ambiguous form (a, b, c) that is an element f ∈ GΔ of order dividing 2 to obtain a coprime factorization of the largest odd divisor of Δ in which Δ = −4ac or Δ = a(a − 4c) or Δ = (b − 2a) (b + 2a). If the ambiguous form provides a factorization of n then stop, otherwise find another ambiguous form until the ...
Trachtenberg system. The Trachtenberg system is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp.