"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."
The largest positive integer that divides two or more integers without a remainder.
Divisibility: The concept of divisibility is central to understanding GCD. Divisibility refers to determining whether a number can be evenly divided by another number without leaving a remainder.
Prime and composite numbers: A prime number is a number greater than 1 that can only be divided evenly by 1 and itself. Composite numbers, on the other hand, can be divided evenly by more than just 1 and themselves.
Factoring: Factoring involves finding an expression that can be multiplied together to get a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Euclid's algorithm: This is a classic algorithm for finding the GCD of two numbers. It involves repeatedly dividing the larger number by the smaller number until the remainder is 0.
Extended Euclidean algorithm: This algorithm is an extension of Euclid's algorithm that allows you to not only find the GCD of two numbers but also the coefficients of the Bezout's identity.
Bezout's identity: Bezout's identity is a theorem that states that for any two integers a, b, there exist integers x and y such that ax + by = gcd(a,b).
Fundamental theorem of arithmetic: This theorem states that every integer greater than 1 can be written as a unique product of prime numbers.
Least common multiple: The least common multiple (LCM) is the smallest number that is a multiple of two or more given numbers.
Modular arithmetic: Modular arithmetic involves working with remainders when dividing by a given number. It has several applications in number theory, including finding the GCD.
Chinese remainder theorem: This theorem states that if two numbers are relatively prime, then any two solutions to a system of congruences are congruent modulo the product of those two numbers.
Euler's totient function: Euler's totient function is a number theory function that counts the positive integers up to a given integer n that are relatively prime to n.
Wilson's theorem: Wilson's theorem is a number theory theorem that states that a number p is prime if and only if (p-1)! + 1 is divisible by p.
Euclidean Algorithm: It is the most common method used to find GCD, based on the division algorithm.
Extended Euclidean Algorithm: Also known as the Bezout's Identity Algorithm, it is used to find a pair of numbers that will satisfy a given linear Diophantine equation.
Binary GCD Algorithm: It is a more efficient algorithm to calculate GCD on large numbers compared to Euclidean Algorithm.
Lehmer's GCD Algorithm: It is a faster algorithm to calculate GCD but it is more complex and requires more memory and space.
Chinese Remainder Theorem: It is a method to calculate GCD for more than two numbers with the help of modular arithmetic.
Stein's algorithm: It is an efficient algorithm to compute GCD of two integers that uses only bit shifting, subtraction, and conditional swaps without multiplication or division.
Linear GCD Algorithm: It is an algorithm that can be used to find the GCD of two numbers without using any division operation, but it can only be used for integers.
Polynomial GCD Algorithm: It is an algorithm that is used to find the greatest common divisor of two polynomials and can be employed for complex numbers.
"For two integers x, y, the greatest common divisor of x and y is denoted gcd(x, y)."
"The GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4."
"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."
"This notion can be extended to polynomials (see Polynomial greatest common divisor)."
"Historically, other names for the same concept have included greatest common measure."
"The greatest common divisor can also be extended to other commutative rings (see ยง In commutative rings below)."
"The paragraph states that the integers should 'not all be zero', so the definition doesn't apply if the dividend and divisor are both zero."
"Yes, the greatest common divisor is the 'largest positive integer that divides each of the integers.'"
"The definition states the 'largest positive integer,' implying that there can only be one greatest common divisor."
"If the greatest common divisor of two integers is 1, it means the two integers are coprime or relatively prime."
"No, the greatest common divisor is a divisor of the integers, so it cannot be larger than them."
"There are no specific restrictions mentioned in the paragraph regarding the values of x and y."
"The paragraph does not mention the possibility of the greatest common divisor being negative, so it is assumed to be positive."
"The definition states that the integers should 'not all be zero,' so the greatest common divisor is undefined in that case."
"The greatest common divisor is the largest positive integer that divides each of the given integers."
"The greatest common divisor is used in various algorithms and mathematical operations involving integers."
"The paragraph specifically refers to the greatest common divisor of integers, so it doesn't apply to non-integer values."
"Yes, the most commonly used notation is gcd(x, y)."
"The paragraph explains that the concept of greatest common divisor can be extended to polynomials and other commutative rings, so the divisor may differ in those contexts."