GCF Calculator (Greatest Common Factor)

The greatest common factor (GCF), also called the greatest common divisor (GCD), of a set of whole numbers is the largest whole number that divides evenly into every number in the set, with no remainder left over. It is one of the most fundamental ideas in number theory and shows up constantly in practical situations, from simplifying fractions to splitting items evenly into groups. The most direct way to find a GCF by hand is to list every factor of each number and pick the largest one they share. Take 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors they share are 1, 2, 3, and 6, and the greatest of those shared factors is 6. So the GCF of 12 and 18 is 6. This listing method works fine for small numbers but becomes slow and error-prone for larger ones — imagine trying to list every factor of 8,904 and 1,491 by hand. That is where the Euclidean algorithm comes in, a far faster method discovered over two thousand years ago and still the standard approach used by calculators and computers today. It relies on a simple but powerful fact: the GCF of two numbers does not change if you replace the larger number with the remainder of dividing it by the smaller number. You repeat this process — divide, take the remainder, replace the larger number with that remainder — until the remainder becomes 0. At that point, the other number left over is the GCF. Let's walk through 8,904 and 1,491 to see how quickly this converges. Divide 8,904 by 1,491: it goes 5 times with a remainder of 8,904 − (5 × 1,491) = 8,904 − 7,455 = 449. Now find the GCF of 1,491 and 449 instead. Divide 1,491 by 449: it goes 3 times with a remainder of 1,491 − 1,347 = 144. Now find the GCF of 449 and 144. Divide 449 by 144: it goes 3 times with a remainder of 449 − 432 = 17. Now find the GCF of 144 and 17. Divide 144 by 17: it goes 8 times with a remainder of 144 − 136 = 8. Now find the GCF of 17 and 8. Divide 17 by 8: it goes 2 times with a remainder of 1. Now find the GCF of 8 and 1. Divide 8 by 1: it goes 8 times exactly, remainder 0. Once the remainder hits 0, the GCF is the last non-zero remainder we were dividing by, which is 1. So 8,904 and 1,491 share no common factor larger than 1 — they are what's called coprime, even though neither is a small number. When you have more than two numbers, the GCF of the whole group is found by applying the Euclidean algorithm pairwise: find the GCF of the first two numbers, then find the GCF of that result and the third number, and so on until every number in the list has been folded in. The order does not matter — GCF is associative, so you get the same final answer regardless of which pair you start with. A common misconception is confusing GCF with LCM (Least Common Multiple), its close relative. GCF is the largest number that divides into all your numbers; LCM is the smallest number that all your numbers divide into. They solve opposite kinds of problems: GCF is for splitting things into equal groups or simplifying ratios, while LCM is for finding when repeating events line up, like syncing schedules. This site's companion LCM Calculator uses the very same Euclidean algorithm under the hood — GCF and LCM are mathematically linked by the identity LCM(a,b) × GCF(a,b) = a × b.