Greatest common divisor
The GCD is sometimes called the greatest common factor (GCF).
A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediately follows that the greatest common divisor of several numbers is divisible by any other common divisor of these numbers.
Finding the GCD
Using prime factorization
Once the prime factorizations of the given numbers have been found, the greatest common divisor is the product of all common factors of the numbers.
and . The common factors are 2 and , so .
The Euclidean algorithm is much faster and can be used to give the GCD of any two numbers without knowing their prime factorizations. To find the greatest common divisor of more than two numbers, one can use the recursive formula .
Using least common multiple
The GCD of two numbers can also be found using the equation , where is the least common multiple of and .