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Greatest common divisor. In mathematics, the greatest common divisor ( GCD) 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, gcd (8, 12) = 4. [ 1][ 2]
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.
Polynomial greatest common divisor. 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 ...
The least common multiple of the denominators of two fractions is the "lowest common denominator" (lcd), and can be used for adding, subtracting or comparing the fractions. The least common multiple of more than two integers a, b, c, . . . , usually denoted by lcm (a, b, c, . . .), is defined as the smallest positive integer that is divisible ...
Definition. Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product of a unit u and zero or more irreducible elements pi of R : x = u p1 p2 ⋅⋅⋅ pn with n ≥ 0.
Bézout's identity. In mathematics, Bézout's identity (also called Bézout's lemma ), named after Étienne Bézout who proved it for polynomials, is the following theorem : Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form ...
Principal ideal domain. In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some authors such as Bourbaki refer to PIDs as principal rings .
In algebra, Gauss's lemma, [ 1] named after Carl Friedrich Gauss, is a theorem [ note 1] about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic ). Gauss's lemma underlies all the theory of factorization ...
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