Greatest common divisor

The greatest common divisor (GCD) of two or more integers is the largest integer that is a divisor of all the given numbers

The GCD is sometimes called the greatest common factor (GCF).

The greatest common divisor of several numbers is divisible by any other common divisor of these numbers.

The GCD can be found in several ways. The first way invlolves factoring both numbers, and the second way uses Euclidean algorithm.

Once the prime factorizations of the given numbers has been found, the greatest common divisor is the product of all common factors of the numbers.

Example: $270=2\times3^3\times5$ $144=2^4\times3^2$

The common factors are 2 and $3^2$ , so the GCD is $2\times3^2=18$

Another Example: $1200=2^4\times3\times5$ $720=2^4\times3^2\times5$ $288=2^5\times3^2$

The common factors are $2^4$ and 3, making the GCD ${2^4\times3=48}$ .

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 $\displaystyle GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)$ .

For two numbers, a and b, the GCD can be at most |a-b|. Also, the GCD can also be no more than the difference in the largest multiple of the smaller number that is less than the larger number. By subtracting the largest multiple of the smaller number from the larger number over and over until the numbers are equal, you will end up with the largest number they both divide, the GCD.

Example: (1440, 560) (1440, 1120) (320, 1120) (960, 1120) (160, 960) (160, 160), so the GCD is 160.

Another Example: (1200, 720, 288) (1200, 720, 576) (624, 144, 576) (624, 576, 576) (624, 576) (48, 576) (48, 48), so the GCD is 48.

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Greatest common divisor