Greatest common divisorIn 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, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc.
Least common multipleIn 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. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0.
Coprime integersIn number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also a is prime to b or a is coprime with b. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor.
Polynomial greatest common divisorIn 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 polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.
Gaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.
Show more