How to Find the Greatest Common Divisor (Two Methods)

The greatest common divisor (GCD) is the largest number that divides two or more numbers evenly. It is widely used when simplifying fractions. This guide shows two ways to find it: listing divisors and using prime factorization.

Method 1: List the divisors

Write out all the divisors (numbers that divide evenly) of each number, then pick the largest one they share.

Let's try 24 and 36. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The shared divisors are 1, 2, 3, 4, 6, 12, and the largest, 12, is the greatest common divisor.

Method 2: Use prime factorization

For larger numbers, prime factorization is faster and more reliable. Break each number into a product of primes, then collect the shared primes at their lowest powers.

24 = 2^3 × 3 and 36 = 2^2 × 3^2. The shared primes are 2 and 3. Take 2 at the lower power (2^2) and 3 at the lower power (3^1). Multiplying gives 2^2 × 3 = 12, the same answer as the listing method.

Three or more numbers, and common mistakes

The idea is the same for three or more numbers. With prime factorization, collect only the primes shared by all of them, each at its lowest power. A common mistake is including a prime that is not shared by every number. Choose only the primes common to all.

Also be careful not to mix up the GCD and the LCM. The GCD is what divides all the numbers, so it is no larger than the original numbers.

What is it used for?

The greatest common divisor is a star player in simplifying fractions. Divide the numerator and denominator by their greatest common divisor to reach the simplest form in one step. Use the prime factorization calculator to break numbers down and the GCD and LCM calculator to check your answer.