GCF Calculator — greatest common factor of two numbers

How it works

The greatest common factor (GCF) — also known as the greatest common divisor (GCD) or highest common factor — is the largest whole number that divides two numbers without leaving a remainder. Every pair of whole numbers has one, and it's at least 1 (which divides everything).

The reliable way to find it is Euclid's algorithm: divide the larger number by the smaller and keep the remainder, then repeat with the smaller number and that remainder, until the remainder reaches 0. The last non-zero remainder is the GCF. It's far faster than listing every factor, especially for large numbers.

Worked example

Find the GCF of 48 and 36. Listing factors, both share 1, 2, 3, 4, 6 and 12 — the largest is 12. By Euclid's algorithm it's quicker:

• 48 ÷ 36 leaves remainder 12
• 36 ÷ 12 leaves remainder 0

The last non-zero remainder is 12. Two numbers with no common factor other than 1, like 17 and 5, are called coprime — their GCF is 1.

Frequently asked questions

How do I find the greatest common factor?

Either list the factors of each number and take the largest they share, or use Euclid's algorithm (repeatedly divide and keep the remainder until it's zero). For small numbers listing is fine; for large ones Euclid's method is much faster and is what this calculator uses.

What's the difference between GCF, GCD and HCF?

Nothing — they're three names for the same thing. **GCF** (greatest common factor), **GCD** (greatest common divisor) and **HCF** (highest common factor) all mean the largest number dividing both inputs. Different textbooks and countries prefer different names.

What does it mean if the GCF is 1?

The two numbers are **coprime** (or relatively prime): they share no common factor except 1, even though neither has to be prime itself. 8 and 15 are coprime, for example. A GCF of 1 means a fraction like 8/15 is already in lowest terms.

How is the GCF used with fractions?

To reduce a fraction to lowest terms, divide the numerator and denominator by their GCF. Since GCF(48, 36) = 12, the fraction 36/48 simplifies to 3/4 in one step. It's the same operation as simplifying a ratio.

How does the GCF relate to the LCM?

They're linked by a neat identity: GCF(a, b) × LCM(a, b) = a × b. So once you have one, the other follows — LCM(a, b) = a × b ÷ GCF(a, b). For 48 and 36, that's 48 × 36 ÷ 12 = 144.

Can I find the GCF of more than two numbers?

Yes — take the GCF of the first two, then the GCF of that result with the next number, and so on. The order doesn't matter. This calculator handles the common two-number case; chain it for three or more.