Search results
Results from the WOW.Com Content Network
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 ...
In mathematics, the Euclidean algorithm, [ note 1 ] or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder.
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer. The GCD of a and b is generally denoted gcd (a, b).
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...
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 ...
hide. In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the ...
where gcd denotes the greatest common divisor of the leading monomials of f and g. As the monomials that are reducible by both f and g are exactly the multiples of lcm, one can deal with all cases of non-uniqueness of the reduction by considering only the S-polynomials. This is a fundamental fact for Gröbner basis theory and all algorithms for ...
The lemma first appeared in Euclid's Elements, and is a fundamental result in elementary number theory. If the premise of the lemma does not hold, that is, if pis a composite number, its consequent may be either true or false. For example, in the case of p= 10, a= 4, b= 15, composite number 10 divides ab= 4 × 15 = 60, but 10 divides neither 4 ...