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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 (,).
The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder. Synonyms for GCD include greatest common factor (GCF), highest common factor (HCF), highest common divisor (HCD), and greatest common measure (GCM).
In this case, the distributive law allows factoring out this common factor. If there are several such common factors, it is preferable to divide out the greatest such common factor. Also, if there are integer coefficients, one may factor out the greatest common divisor of these coefficients.
m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor). lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and ...
Any two elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d that divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. Any UFD is integrally closed.
Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field.
Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored using general-purpose techniques. The largest factor found using ECM so far has 83 decimal digits and was discovered on 7 September 2013 by R. Propper. [1]
The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right ...
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