This GCF calculator finds the greatest common factor of two or more whole numbers — the largest number that divides every input with no remainder. It uses the Euclidean algorithm, the same method taught in number theory courses, and also shows each number’s complete factor list, the LCM, and whether the numbers are coprime.
The GCF (also called the GCD, greatest common divisor, or HCF, highest common factor) is what you use to reduce fractions to lowest terms, split quantities into the largest equal groups, and simplify ratios. For example, knowing that GCF(48, 60) = 12 tells you instantly that 48/60 reduces to 4/5.
The Euclidean Algorithm, Explained
The Euclidean algorithm finds the GCF of two numbers without factoring either one. It relies on a single fact: the GCF of a and b is the same as the GCF of b and the remainder of a ÷ b. Repeat until the remainder is zero — the last nonzero remainder is the GCF.
Example: GCF(270, 192).
- 270 ÷ 192 = 1 remainder 78
- 192 ÷ 78 = 2 remainder 36
- 78 ÷ 36 = 2 remainder 6
- 36 ÷ 6 = 6 remainder 0
The last nonzero remainder is 6, so GCF(270, 192) = 6. For more than two numbers, apply the algorithm pairwise: GCF(a, b, c) = GCF(GCF(a, b), c). The method is extremely fast even for numbers with hundreds of digits.
GCF vs LCM: What Each One Means
The GCF and LCM answer opposite questions:
- GCF: the largest number that divides into all of the inputs. It is never bigger than the smallest input. Use it to simplify fractions and split things into the largest possible equal groups.
- LCM: the smallest number that all of the inputs divide into. It is never smaller than the largest input. Use it to find common denominators and align repeating cycles.
For exactly two numbers they are linked by the identity GCF(a, b) × LCM(a, b) = a × b. Take 48 and 60: the GCF is 12 and the LCM is 240, and indeed 12 × 240 = 2,880 = 48 × 60. If the GCF is 1, the numbers are called coprime and the LCM is simply their product.
Worked Example: GCF of 48, 60, and 72
Find the GCF of 48, 60, and 72 by listing factors:
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The common factors are 1, 2, 3, 4, 6, and 12 — so the GCF is 12.
The Euclidean route is faster: GCF(48, 60) → 60 = 1×48 + 12, then 48 = 4×12 + 0, giving 12. Then GCF(12, 72) = 12, since 72 = 6×12 exactly. A practical read: 48, 60, and 72 items can each be split into full boxes of 12, and 12 is the largest box size that works for all three.
Frequently Asked Questions
What is the greatest common factor (GCF)?
The GCF of a set of whole numbers is the largest number that divides every one of them with no remainder. For example, the GCF of 18 and 24 is 6, because 6 is the biggest number appearing in both factor lists. It is also called the GCD (greatest common divisor) or HCF (highest common factor).
Is GCF the same as GCD?
Yes. GCF (greatest common factor), GCD (greatest common divisor), and HCF (highest common factor) are three names for exactly the same value. American schools typically say GCF, mathematicians and computer scientists usually say GCD, and British curricula often use HCF.
How does the Euclidean algorithm find the GCF?
Divide the larger number by the smaller and keep the remainder. Then divide the previous divisor by that remainder, and repeat until the remainder is 0 — the last nonzero remainder is the GCF. For 48 and 60: 60 ÷ 48 leaves 12, then 48 ÷ 12 leaves 0, so the GCF is 12.
What does it mean if two numbers are coprime?
Two numbers are coprime (or relatively prime) when their GCF is 1 — they share no common factor besides 1. For example, 8 and 15 are coprime even though neither is prime. When numbers are coprime, their LCM equals their product, and a fraction made from them is already in lowest terms.
How do I use the GCF to simplify a fraction?
Divide both the numerator and the denominator by their GCF. For 48/60, the GCF is 12, so divide top and bottom by 12 to get 4/5, which is fully reduced. Using the GCF guarantees one-step simplification — dividing by any smaller common factor would leave more reducing to do.