Arithmetic
GCF Calculator
Find the greatest common factor of two or more whole numbers — the largest integer that divides every number with no remainder. Three methods, every step shown, plus a GCF lookup chart for common pairs.
Enter your numbers
3 numbersTwo or more whole numbers, separated by commas or spaces.
Reading: 12, 18, 30
Greatest common factor
6
The largest whole number that divides 12, 18, and 30 evenly.
Each number ÷ 6 is a whole number
Shared prime factors
2 × 3 = 6
Euclidean reduction
- GCF(12, 18) = 6 · 2 steps
- GCF(6, 30) = 6 · 1 step
LCM
180
Numbers
3
GCF × LCM = product of the numbers (for two inputs). Negatives use their absolute value.
Try an example
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The basics
What is the greatest common factor?
The greatest common factor (GCF) of a set of whole numbers is the biggest number that divides all of them with nothing left over. It is the cornerstone of reducing fractions, splitting things into equal groups, and simplifying ratios — anywhere you need the largest shared piece.
largest d : a ÷ d, b ÷ d whole
The largest shared divisor
List the factors each number has in common, and the GCF is simply the biggest one on that shared list. Every common factor divides the GCF too.
12 → 1,2,3,4,6,12 · 18 → 1,2,3,6,9,18 · shared 1,2,3,6 → GCF 6
GCF = GCD = HCF
One number, three names
Greatest common factor, greatest common divisor, and highest common factor all mean the same thing. The label changes by textbook and country; the math never does.
Method
How to find the GCF by hand
The Euclidean algorithm is the fastest reliable method — it needs no factoring and never stalls, even on huge numbers. Four moves, repeated until the remainder is zero.
- 1
Put the larger number first
Take the two numbers and call the bigger one a and the smaller one b. The algorithm only ever shrinks them.
a = 18, b = 12 - 2
Divide and keep the remainder
Divide a by b and note the remainder r. Ignore the quotient — only the remainder matters.
18 = 1 × 12 + 6 → r = 6 - 3
Shift down and repeat
The old b becomes the new a, and the remainder becomes the new b. Divide again.
12 = 2 × 6 + 0 → r = 0 - 4
Stop at remainder zero
When the remainder hits 0, the last divisor you used is the GCF. For a list, fold that answer into the next number and repeat.
GCF(18, 12) = 6
Worked examples
Three ways to the same answer
Listing factors, prime factorization, and the Euclidean algorithm all land on the same GCF. Watch each one solve GCF(24, 36) — pick whichever fits the numbers in front of you.
Find GCF(24, 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: 1, 2, 3, 4, 6, 12
Greatest of those: 12
GCF(24, 36) = 12
Where it shows up
Why the GCF matters
Far from a classroom curiosity, the GCF is the quiet workhorse behind clean fractions, fair groupings, and tidy ratios.
num ÷ GCF / den ÷ GCF
Simplifying fractions
Dividing the top and bottom of a fraction by their GCF reduces it to lowest terms in a single step instead of halving over and over.
18/24 ÷ 6 = 3/4
items ÷ GCF per group
Equal groups & packing
When you must split two quantities into identical bundles with nothing left over, the GCF is the largest bundle count that works.
GCF(24, 36) = 12 → 12 kits
a : b ÷ GCF
Reducing ratios
A ratio simplifies exactly like a fraction — divide both sides by their GCF to express it in its smallest whole-number form.
20 : 30 ÷ 10 = 2 : 3
Reference
Factors of common numbers
The building blocks the GCF draws on. Every common factor of two numbers is a factor each one already has — so the GCF lives in the overlap of these lists.
| Number | Prime factorization | All factors |
|---|---|---|
| 6 | 2 × 3 | 1, 2, 3, 6 |
| 8 | 2³ | 1, 2, 4, 8 |
| 9 | 3² | 1, 3, 9 |
| 12 | 2² × 3 | 1, 2, 3, 4, 6, 12 |
| 16 | 2⁴ | 1, 2, 4, 8, 16 |
| 18 | 2 × 3² | 1, 2, 3, 6, 9, 18 |
| 20 | 2² × 5 | 1, 2, 4, 5, 10, 20 |
| 24 | 2³ × 3 | 1, 2, 3, 4, 6, 8, 12, 24 |
| 30 | 2 × 3 × 5 | 1, 2, 3, 5, 6, 10, 15, 30 |
| 36 | 2² × 3² | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 48 | 2⁴ × 3 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| 60 | 2² × 3 × 5 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
Properties
Rules worth remembering
A handful of facts make GCF problems quicker — and let you sanity-check an answer at a glance.
GCF(a, b) = GCF(b, a)
Order doesn't matter
Swapping the inputs never changes the GCF — it depends only on the set of numbers, not their arrangement.
GCF(a, b) ≤ min(a, b)
Never bigger than the smallest
The GCF can't exceed the smallest number in the set, because nothing larger than b can divide b.
b mod a = 0 → GCF = a
When one divides the other
If the smaller number divides the larger exactly, the smaller number is the GCF. There is nothing left to reduce.
GCF = 1 → coprime
Coprime numbers
A GCF of 1 means the numbers share no factor beyond 1. They need not be prime — 8 and 15 are coprime, yet neither is prime.
GCF(a, b) × LCM(a, b) = a × b
The GCF–LCM link
For any two numbers, multiply the GCF by the LCM and you get the product of the numbers — a fast way to find one from the other.
GCF(ka, kb) = k · GCF(a, b)
Scales with a common factor
Multiply both numbers by the same value and the GCF scales by that value too — handy when clearing decimals or units.
Worth knowing
GCF and LCM are two sides of a coin
The greatest common factor and the least common multiple are bound together. Knowing one instantly gives you the other for a pair of numbers.
GCF — the shared floor
Largest shared divisor
The GCF reaches down to the biggest number both share as a factor. It is at most the smaller input.
GCF(12, 18) = 6
LCM — the shared ceiling
Smallest shared multiple
The LCM reaches up to the smallest number both divide into. It is at least the larger input. And 6 × 36 = 12 × 18 = 216.
LCM(12, 18) = 36
Watch out
Common mistakes
The errors that turn an easy GCF into a wrong answer.
GCF ≠ LCM
Mixing up GCF and LCM
Reaching for the smallest shared multiple when the question wants the largest shared factor. GCF goes down to the overlap; LCM goes up.
use min power, not max
Taking the wrong prime power
With prime factorization you take the lowest power of each shared prime, not the highest — the highest gives the LCM instead.
stop at a common factor
Settling for a common factor
Finding 2 divides both and stopping. Keep going until no larger shared factor exists — that is what makes it the greatest.
GCF(n, 0) = n
Forgetting the zero rule
The GCF of any number and 0 is that number, since every integer divides 0. A bare 0 in the list shouldn't drag the answer to 0.
Quick chart
What is the GCF of X and Y?
Type any two numbers for an instant answer, or scan the most-searched pairs below. Each row is the worked GCF of a common pairing.
GCF ofand=6
Both 12 and 18 are divisible by 6, and by no larger number.
Tap any pair to load it into the lookup above.
FAQ
Common questions
The greatest common factor of two or more whole numbers is the largest positive integer that divides each of them with no remainder. For 12 and 18 it is 6, because 6 divides both and no larger number does.
Yes. Greatest common factor (GCF), greatest common divisor (GCD), and highest common factor (HCF) are three names for exactly the same number. Different textbooks and countries prefer different labels, but the calculation is identical.
Divide the larger number by the smaller and keep the remainder. Replace the larger number with the smaller, and the smaller with the remainder, then repeat. When the remainder reaches 0, the last non-zero divisor is the GCF. For 18 and 12: 18 = 1×12 + 6, then 12 = 2×6 + 0, so the GCF is 6.
If the numbers have no common factor other than 1, their GCF is 1 and they are called coprime, or relatively prime. For example, the GCF of 8 and 15 is 1 even though neither is prime.
Yes. Take the GCF of the first two numbers, then find the GCF of that result with the third number, and so on. Because GCF(a, b, c) = GCF(GCF(a, b), c), folding the running answer into each next number gives the GCF of the whole list.
The GCF is the largest number that divides all the inputs, while the LCM (least common multiple) is the smallest number that all the inputs divide into. The GCF is never larger than the smallest input; the LCM is never smaller than the largest input. For two numbers, GCF × LCM equals their product.
Divide both the numerator and denominator by their GCF. 18/24 has a GCF of 6, so dividing both by 6 gives 3/4 — the fraction in lowest terms. Reducing in one step instead of repeatedly halving is the GCF's most common everyday job.
The GCF is defined for whole numbers, so it uses the absolute value of any negative input and the answer is always positive. Decimals have no GCF directly — scale them to whole numbers first (for example multiply 0.6 and 0.9 by 10 to get 6 and 9).
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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