GCF Calculator
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About the GCF
The greatest common factor answers “what's the biggest piece that measures both?” — the largest tile that fits two wall dimensions evenly, the biggest group size that splits two class counts with nobody left over, the number that reduces a fraction in one step. Same concept, three names: GCF, GCD (mathematicians), HCF (UK classrooms).
Enter two or more whole numbers. You get the GCF, each number expressed as GCF × cofactor, the Euclid steps, and the LCM alongside — because the two travel together in fraction work.
The other half of the pair has its own tool — the LCM Calculator
Euclid's Algorithm
The oldest algorithm still in daily use (~300 BC), and still the fastest way by hand:
GCF(a, b) = GCF(b, a mod b), repeat until the remainder is 0 The last nonzero remainder is the GCF Multiple numbers: GCF(a, b, c) = GCF(GCF(a, b), c)
Worked example: GCF(48, 18) → 48 = 2×18 + 12 → 18 = 1×12 + 6 → 12 = 2×6 + 0 → GCF is 6. Three divisions instead of listing every factor of both numbers — the efficiency is why your computer uses the same method inside fraction libraries.
Worked Examples
Common cases — each computed by this calculator:
| Numbers | GCF | Reduced by it | LCM |
|---|---|---|---|
| 12, 18 | 6 | 2, 3 | 36 |
| 24, 36 | 12 | 2, 3 | 72 |
| 15, 25 | 5 | 3, 5 | 75 |
| 48, 18 | 6 | 8, 3 | 144 |
| 24, 36, 60 | 12 | 2, 3, 5 | 360 |
| 17, 31 | 1 (coprime) | 17, 31 | 527 |
GCF = 1 means the numbers are coprime — they share no factors, and their fraction is already in lowest terms.
Where GCF Shows Up
Fractions are the classic customer: dividing numerator and denominator by their GCF reduces to lowest terms in one move (18/24 ÷ 6 = 3/4). Ratios simplify identically — a 24:36 mix is really 2:3. It's the same operation our fraction calculator performs on every answer.
Real-world sizing problems are secretly GCF questions: the largest square tile that fits a 48 × 18 area exactly (6 units), cutting two board lengths into equal pieces with no waste, splitting 24 pencils and 36 erasers into identical kits (12 kits). Whenever the question is “biggest equal division of several quantities,” it's this calculator.
Frequently Asked Questions
What is the GCF of 12 and 18?
6 — the largest number dividing both (12 = 6×2, 18 = 6×3). Euclid gets there in two steps: 18 = 1×12 + 6, then 12 = 2×6 + 0.
Are GCF, GCD, and HCF the same thing?
Yes — greatest common factor, greatest common divisor, highest common factor. US schools say GCF, mathematicians GCD, UK classrooms HCF. One concept, one algorithm.
How do I find the GCF by hand?
Euclid's algorithm beats factor-listing: divide the larger by the smaller, keep the remainder, repeat with (smaller, remainder) until remainder 0 — the last divisor is the GCF. For 48 and 18 it takes three quick divisions.
What does GCF = 1 mean?
The numbers are coprime (relatively prime) — no shared factors. Any fraction made from them is already fully reduced, and their LCM is simply their product. Note coprime doesn't require either number to be prime: 8 and 9 are coprime.
How does GCF relate to LCM?
For two numbers: GCF × LCM = a × b, always. So knowing one gives the other for free — this calculator shows both, and the identity is a handy self-check on homework.
How do I reduce a fraction with the GCF?
Divide top and bottom by it: 24/36 has GCF 12, giving 2/3 in one step — guaranteed lowest terms, no repeated halving. That single division is the entire art of simplifying fractions.
Methodology. This calculator uses standard, peer-reviewed mathematical formulas. It is reviewed and maintained by the Vast Calculators editorial team.
Last updated · July 11, 2026
Results are estimates for general use; verify critical figures independently.