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Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is mainly used for big integers that have a representation as a string of digits relative to some chosen numeral system base , say β = 1000 or β = 2 32 .
Factoring out the gcd's from the coefficients, we can write = ′ and = ′ where the gcds of the coefficients of ′, ′ are both 1. Clearly, it is enough to prove the assertion when f , g {\displaystyle f,g} are replaced by f ′ , g ′ {\displaystyle f',g'} ; thus, we assume the gcd's of the coefficients of f , g {\displaystyle f,g} are ...
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2.
As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers a, b is a Gaussian integer d that is a common divisor of a and b, which has all common divisors of a and b as divisor. That is (where | denotes the divisibility relation), d | a and d | b, and; c | a and c | b implies c | d.
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that + = (,).
Suppose G is a finite group of order n, and d is a divisor of n.The number of order d elements in G is a multiple of φ(d) (possibly zero), where φ is Euler's totient function, giving the number of positive integers no larger than d and coprime to it.
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The previous result says that a solution exists if and only if gcd(a, m) = 1, that is, a and m must be relatively prime (i.e. coprime). Furthermore, when this condition holds, there is exactly one solution, i.e., when it exists, a modular multiplicative inverse is unique: [ 8 ] If b and b' are both modular multiplicative inverses of a respect ...