Greatest Common Divisor

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. For two integers x, y, the greatest common divisor of x and y is denoted gcd(x,y). For example, the GCD of 8 and 12 is 4, that is, gcd(8,12)=4.

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. Historically, other names for the same concept have included greatest common measure.


The number 54 can be expressed as a product of two integers in several different ways: 54 x 1 = 27 × 2 = 18 × 3 = 9 × 6.

Thus the complete list of divisors of 54 is: 1,2,3,6,9,18,27,54. Similarly, the divisors of 24 are: 1,2,3,4,6,8,12,24. The numbers that these two lists have in common are the common divisors of 54 and 24, that is: 1,2,3,6. Of these, the greatest is 6, so it is the greatest common divisor: gcd(54,24)=6.

Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient methods exist, such as, Euclid's Algorithm or Prime Factorization

This page was last updated: 2023-04-11 Tue 15:22. Source