How to Find the GCF

Method 1: Listing factors

This is the most direct method and the easiest to understand for small numbers, which is why it's usually taught first. List every factor of each number (every whole number that divides it evenly), then find the factors that appear in every list, and pick the largest one. You can check your answer any time on the GCF calculator, or look up a small pair directly on the GCF reference chart.

Worked example: find the GCF of 24 and 36.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors (appear in both lists): 1, 2, 3, 4, 6, 12
• Greatest common factor: 12, the largest number in that shared list

When listing factors gets impractical

Listing factors works well for numbers you can factor in your head, roughly up to two-digit numbers, but it breaks down fast for anything larger. Finding every factor of a number like 840 or 1,260 by hand means checking dozens of possible divisors, and it's easy to accidentally miss one, which silently gives you a wrong (too small) GCF. For numbers that size, prime factorization or the division method below are both faster and less error-prone, because neither one requires you to find and list every single factor first.

Method 2: Prime factorization

Break every number down into a product of prime numbers (numbers only divisible by 1 and themselves, like 2, 3, 5, 7, 11...). Then, for each prime that appears in every number's factorization, take the lowest power it appears with, and multiply those together. That product is the GCF.

Worked example: find the GCF of 60 and 84.

• Prime factorization of 60: 2 × 2 × 3 × 5 = 2² × 3 × 5
• Prime factorization of 84: 2 × 2 × 3 × 7 = 2² × 3 × 7
• Shared primes: both numbers include 2² and a single factor of 3 (60's factor of 5 and 84's factor of 7 are not shared, so they're excluded)
• Multiply the shared primes at their lowest shared power: 2² × 3 = 4 × 3 = 12, so the GCF of 60 and 84 is 12

Why prime factorization is worth learning even though it's more steps

Prime factorization takes an extra step compared to the division method below, but it has a real advantage: it shows you why the answer is what it is, in a way you can double-check visually, and it's the same technique you need for finding the LCM (multiply every prime that appears anywhere, at its highest power, the mirror image of the GCF rule). If a problem asks for both the GCF and the LCM of the same numbers, doing the prime factorization once and reading off both answers is often faster overall than two separate calculations.

Method 3: The division method (Euclidean algorithm)

This is the fastest method for large numbers, and it's the method this site's calculator uses internally. Divide the larger number by the smaller number and note the remainder. Then divide the previous smaller number by that remainder. Keep repeating, replacing the pair of numbers each time, until you reach a remainder of 0. The last non-zero remainder is the GCF.

Worked example: find the GCF of 252 and 105, numbers large enough that listing every factor would be tedious.

• 252 ÷ 105 = 2 remainder 42 (105 × 2 = 210, and 252 − 210 = 42)
• 105 ÷ 42 = 2 remainder 21 (42 × 2 = 84, and 105 − 84 = 21)
• 42 ÷ 21 = 2 remainder 0
• The remainder just hit 0, so the previous remainder, 21, is the GCF of 252 and 105

Which method should you actually use?

For small numbers you can factor in your head (roughly anything under 50), listing factors is fine and often fastest since there's no setup. For medium numbers, or when you also need the LCM, prime factorization is usually the clearer choice because it's easy to check by eye. For large numbers, or any time you want a guaranteed-fast method that never requires finding a full factor list, use the division (Euclidean) method, or simply use the calculator above, which runs the same Euclidean method and shows the prime factorization as well, so you get the speed of one method and the visual check of the other in a single result.