Greatest Common Divisor Calculator

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GCD Calculator

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The greatest common divisor

Here are some important points about GCD:

Definition:
The GCD of two or more integers is the largest positive integer that divides each of the given integers without leaving a remainder.

The GCD of two integers a and b is denoted as GCD(a,b).

Properties:
1. GCD(a,b) = GCD(b,a) (commutative property)
2. GCD(a,b) = GCD(-a,b) = GCD(a,-b) = GCD(-a,-b) (absolute value property)
3. If a is a multiple of b, then GCD(a,b) = b
4. If a and b are relatively prime (i.e., they have no common factors except 1), then GCD(a,b) = 1
Calculation:
There are several methods for calculating the GCD of two integers, including the Euclidean algorithm, prime factorization method, and continued fractions method.
Applications:
The GCD has many applications in mathematics, including simplifying fractions, finding equivalent fractions, solving linear Diophantine equations, and cryptography.
Extended GCD:
The extended GCD algorithm is used to find the coefficients of Bézout's identity, which states that the GCD of two integers a and b can be expressed as a linear combination of a and b.

The extended GCD algorithm is also used in modular arithmetic and cryptography.