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On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300). 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, the largest number that divides them both without a remainder.
Animation showing an application of the Euclidean algorithm to find the greatest common divisor of 62 and 36, which is 2. A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclidean division (also called division with remainder) of a by b.
A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. A particular Euclidean function f is not part of the definition of a Euclidean domain, as, in general, a Euclidean domain may admit many different Euclidean functions. In this context, q and r are called respectively a quotient and a remainder of ...
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 to m. Outer loop: Iterate until one of a or b is zero: Decrease m by one.
A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem. If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials (s, t) such that
A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: Input A set of polynomials F that generates I Output A Gröbner basis G for I. G := F; For every f i, f j in G, denote by g i the leading term of f i with respect to the given monomial ordering, and by a ij the least common multiple of g ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Euclidean domains (also known as Euclidean rings) [7] are defined as integral domains which support the following generalization of Euclidean division: Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d (also known as a Euclidean valuation [ 8 ] or degree function [ 7 ] ), there exist q and ...