Greatest Common Factor (GCF / GCD) Calculator

Enter 2 or more whole numbers separated by commas (,). You can find the GCF of 3 or more at once, and the prime factorization of each number and their common prime factors are shown too.

Whole numbers (separated by commas)

Enter only whole numbers of 1 or more, separated by commas (for example, 12, 18). 0, negative numbers and decimals cannot be used.

Result

Enter whole numbers separated by commas in the field on the left and press "Calculate". The result will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter whole numbers separated by commas and get the greatest common factor (GCF, also called the GCD) on the spot
Not just 2 numbers: find the GCF of 3 or more at once, such as "16, 88, 104"
The prime factorization of each number and the prime factors they share are shown too, so you also learn how to find and check the answer
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

Only whole numbers of 1 or more (positive whole numbers) can be used. This page does not handle 0, negative numbers or decimals.

What is this calculation used for?

Simplifying fractions (its most familiar use)

To simplify \(\frac{12}{18}\), divide the numerator and the denominator by their GCF, 6, and you get \(\frac{2}{3}\) in one step.
Instead of dividing by 2, then by 3 and so on, the GCF takes the fraction to simplest form at once. From elementary school math to everyday work, this is the most common use of the GCF.

Splitting things into equal groups with nothing left over (parties and handouts)

You want to pack 24 cookies and 36 juice boxes into as many identical bags as possible, with nothing left over. The largest number of bags is \(\gcd(24, 36) = 12\), with 2 cookies and 3 juice boxes in each bag.
Making prize bags for a school fair or putting supplies into matching kits: this is the standard calculation for splitting different kinds of items into equal groups.

Laying tiles and cutting paper (DIY and design)

To cover a floor that is 84 inches by 108 inches (7 ft by 9 ft) with square tiles, with no gaps and no cutting, the largest tile you can use is \(\gcd(84, 108) = 12\), a 12-inch square tile (7 rows of 9 tiles).
Dividing a rectangle into the largest possible squares is used when tiling floors and walls, and when cutting cards of the same size from a large sheet of paper without waste.

Designing gears (machines and clocks)

If the tooth counts of two meshing gears have a large GCF, the same teeth hit each other again and again, and they wear unevenly. So in machine design, a basic rule is to make the tooth counts as close to relatively prime (GCF of 1) as possible.
The gears inside cars and clocks use this idea of keeping the GCF small.

The encryption that keeps the internet safe

RSA encryption, used for online shopping and online banking, needs to check whether two numbers are relatively prime while making its keys, and the Euclidean algorithm does that job.
A method for the GCF from more than 2,000 years ago now protects communication all over the world. It shows how long-lived mathematics can be.

Formula

Using prime factorization (the basic method)

Standard notation (the usual math form)

\(\gcd(a,\ b)\) \(=\) \(p_1 \times p_2 \times \cdots\)

In words (symbols replaced with words)

② \(\gcd(a, b)\): GCF of \(a\) and \(b\) \(=\) ① product of all the common prime factors

The formula in words

① Write both numbers as products of primes and multiply all the \(p_1, p_2, \ldots\): common prime factors

② to get the \(\gcd(a, b)\): greatest common factor

Quick example

The GCF of 12 and 18 (12 = 2 × 2 × 3 and 18 = 2 × 3 × 3, so they share one 2 and one 3) is

GCF of 12 and 18 \(=\) product of the common prime factors (2 and 3)

\(12 = 2 \times 2 \times 3,\quad 18 = 2 \times 3 \times 3\)

\(\gcd(12,\ 18) = 2 \times 3 = 6\)

Key idea

The prime factors of a whole number are the primes you get when you break it into a product of primes only. If the same prime appears several times in both numbers, use it as many times as it appears in the number with fewer copies. For example, 8 = 2 × 2 × 2 and 12 = 2 × 2 × 3: 2 appears 3 times in 8 and 2 times in 12, so they share it 2 times. So \(\gcd(8, 12) = 2 \times 2 = 4\).

The Euclidean algorithm (the method for large numbers)

Standard notation (the usual math form)

\(\gcd(a,\ b)\) \(=\) \(\gcd(b,\ a \bmod b)\)

In words (symbols replaced with words)

② \(\gcd(a, b)\): GCF of \(a\) and \(b\) \(=\) ① GCF of "the smaller number \(b\)" and "the remainder of \(a \div b\)"

The formula in words

① Divide the larger number by the smaller one, and replace the pair with the pair "smaller number \(b\)" and "remainder \(a \bmod b\)"

② and the \(\gcd(a, b)\): greatest common factor stays the same. The numbers keep getting smaller, and when the remainder becomes 0, the number you divided by is the GCF

Quick example

The GCF of 48 and 18 (48 ÷ 18 = 2 R12) is

GCF of 48 and 18 \(=\) GCF of "18" and "remainder 12"

\(48 = 18 \times 2 + 12\)

\(18 = 12 \times 1 + 6\)

\(12 = 6 \times 2 + 0\)

\(\gcd(48,\ 18) = 6\)

Key idea

Prime factorization suddenly gets hard as the numbers grow (try factoring a number like 134221: it is actually 79 × 1699). With the Euclidean algorithm, you only repeat "divide and take the remainder", and it always reaches the GCF, however large the numbers are. The method appears in Euclid's "Elements" from around the 3rd century BC and is often called the oldest algorithm in the world.

GCF of 3 or more numbers

Standard notation (the usual math form)

\(\gcd(a,\ b,\ c)\) \(=\) \(\gcd(\) \(\gcd(a,\ b)\) \(,\) \(c\) \()\)

In words (symbols replaced with words)

③ \(\gcd(a, b, c)\): GCF of the three numbers \(=\) \(\gcd(\) ① GCF of \(a\) and \(b\), found first \(,\) ② \(c\): the remaining number \()\)

The formula in words

① First find the \(\gcd(a, b)\): GCF of two of the numbers

② then find the GCF of that result and the \(c\): the remaining number

③ to get the \(\gcd(a, b, c)\): GCF of the three numbers

Quick example

The GCF of 16, 88 and 104 (first, the GCF of 16 and 88 is 8) is

GCF of 16, 88 and 104 \(=\) \(\gcd(\) GCF of 16 and 88 (8) \(,\) remaining number (104) \()\)

\(\gcd(16,\ 88) = 8\)

\(\gcd(8,\ 104) = 8\)

\(\gcd(16,\ 88,\ 104) = 8\)

Key idea

However many whole numbers there are, finding the GCF two at a time gives the GCF of all of them. It works the same for 4 or more numbers, and changing the order does not change the answer. The calculator on this page also works through the numbers two at a time, exactly as in this formula.

How the GCF is related to the LCM

Standard notation (the usual math form)

\(\gcd(a,\ b)\) \(\times\) \(\mathrm{lcm}(a,\ b)\) \(=\) \(a \times b\)

In words (symbols replaced with words)

① \(\gcd(a, b)\): greatest common factor \(\times\) ② \(\mathrm{lcm}(a, b)\): least common multiple \(=\) ③ \(a \times b\): product of the two numbers

The formula in words

① Multiply the \(\gcd(a, b)\): greatest common factor

② by the \(\mathrm{lcm}(a, b)\): least common multiple

③ and you always get the \(a \times b\): product of the two numbers

Quick example

For 12 and 18 (GCF 6, LCM 36):

GCF (6) \(\times\) LCM (36) \(=\) product of the two numbers (12 × 18)

\(6 \times 36 = 216\)

\(12 \times 18 = 216\)

Key idea

With this relationship, once you know the GCF you can find the LCM as \(\mathrm{lcm}(a, b) = a \times b \div \gcd(a, b)\) (for large numbers this is more reliable than listing multiples). Note that this relationship holds as is only for two numbers. For 3 or more numbers it does not hold in general.

The greatest common factor is the largest positive whole number that divides every one of the numbers evenly. For small numbers, multiply the prime factors they share. For large numbers, use the Euclidean algorithm, which keeps replacing the pair with the remainder.

Symbols and terms

Symbols

\(\gcd(a, b)\)	G C D of a and b	The greatest common factor of \(a\) and \(b\). The symbol comes from "greatest common divisor" (GCD). US schools usually say greatest common factor (GCF) and may write \(\mathrm{GCF}(a, b)\), which is exactly the same thing.
\(a \bmod b\)	a mod b	The remainder when \(a\) is divided by \(b\). (Example - \(48 \bmod 18 = 12\), because 48 ÷ 18 = 2 R12)
\(\mathrm{lcm}(a, b)\)	L C M of a and b	The least common multiple of \(a\) and \(b\), from the first letters of "least common multiple". (Example - \(\mathrm{lcm}(12, 18) = 36\))
\(p_1, p_2, \ldots\)	p sub 1, p sub 2, and so on	The common prime factors listed in order. The small lowered numbers (subscripts) only show first, second and so on. They are not used in the calculation.
\(\cdots\)	dots (ellipsis)	A symbol for "and so on, following the same pattern". Here it shows that you multiply the prime factors in the same way however many there are.

Terms

factor	A positive whole number that divides a whole number evenly. 12 has 6 factors - 1, 2, 3, 4, 6 and 12.
common factor	A factor shared by two or more whole numbers. The common factors of 12 and 18 are 1, 2, 3 and 6. Every common factor is a factor of the greatest common factor (6 here).
greatest common factor (GCF)	The largest of the common factors. US schools call it the greatest common factor (GCF). In higher math and programming it is usually called the greatest common divisor (GCD). Both are the same number.
prime number	A whole number of 2 or more whose only factors are 1 and itself - 2, 3, 5, 7, 11, 13, … 1 is not a prime number.
prime factorization	Writing a whole number as a product of prime numbers only. (Example - \(12 = 2 \times 2 \times 3\)) Each prime used is called a prime factor.
relatively prime	When the GCF of two whole numbers is 1 (their only common factor is 1). Like 17 and 13, they share no prime factors at all. Also called coprime.
Euclidean algorithm	A way to find the GCF by repeating "divide the larger number by the smaller one, and replace the pair with the smaller number and the remainder". It is taught in high school and college number theory and computer science. Known since ancient times, it is often called the oldest algorithm in the world.
least common multiple (LCM)	The smallest positive whole number that is a multiple of two or more whole numbers. It is used to find a common denominator for fractions. For two numbers, \(\gcd(a, b) \times \mathrm{lcm}(a, b) = a \times b\).

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Multiplication facts and division with remainders (Grades 3–4)	Being able to tell "divides evenly" from "has a remainder" Being able to do division with remainders, such as 48 ÷ 18 = 2 R12
Factors and common factors (Grades 4–6)	Being able to list all the factors of 12 - 1, 2, 3, 4, 6, 12 Being able to find the factors two numbers share (common factors)
Simplifying fractions (Grades 4–5)	Knowing that dividing the numerator and the denominator by the same number does not change the value of a fraction
Prime numbers and prime factorization (Grades 4–6)	Being able to recognize prime numbers (numbers with no factors other than 1 and themselves) Being able to break a whole number into a product of primes, as in \(12 = 2 \times 2 \times 3\)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the GCF of two numbers

First number a	12
Second number b	18
GCF gcd(a, b)	=GCD(B1,B2)

Table to find the GCF of 3 or more numbers

Number 1	16
Number 2	88
Number 3	104
GCF	=GCD(B1:B3)

Table to check the Euclidean algorithm

Larger number a	48
Smaller number b	18
Remainder of a ÷ b (a mod b)	=MOD(B1,B2)
gcd(a, b)	=GCD(B1,B2)
gcd(b, remainder) (same as above)	=GCD(B2,B3)

Table to check GCF × LCM = product of the two numbers

First number a	12
Second number b	18
GCF gcd(a, b)	=GCD(B1,B2)
LCM lcm(a, b)	=LCM(B1,B2)
gcd × lcm	=B3*B4
a × b (same as above)	=B1*B2

Excel has a built-in GCD function for the greatest common factor. "=GCD(B1,B2)" gives the GCF of the values in B1 and B2, so the first table shows 6 in B3.
With a range, as in "=GCD(B1:B3)" in the second table, you can find the GCF of 3 or more numbers at once (B4 shows 8).
The third table checks the Euclidean algorithm. With the MOD function (the remainder), you can confirm that gcd(a, b) and gcd(b, remainder) are the same value (both 6).
In the fourth table, gcd × lcm and a × b both come to 216, which confirms the relationship with the LCM. Just replace the input numbers with your own.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the GCF of two numbers

First number a	12
Second number b	18
GCF gcd(a, b)	=GCD(B1,B2)

Table to find the GCF of 3 or more numbers

Number 1	16
Number 2	88
Number 3	104
GCF	=GCD(B1:B3)

Table to check the Euclidean algorithm

Larger number a	48
Smaller number b	18
Remainder of a ÷ b (a mod b)	=MOD(B1,B2)
gcd(a, b)	=GCD(B1,B2)
gcd(b, remainder) (same as above)	=GCD(B2,B3)

Table to check GCF × LCM = product of the two numbers

First number a	12
Second number b	18
GCF gcd(a, b)	=GCD(B1,B2)
LCM lcm(a, b)	=LCM(B1,B2)
gcd × lcm	=B3*B4
a × b (same as above)	=B1*B2

Google Sheets has GCD, LCM and MOD functions with the same names as Excel, so the same formulas work as is.
Copy the whole table, paste it into cell A1, and replace the input numbers with your own.

How to calculate it in Python

from math import gcd
from functools import reduce

numbers = [330, 75, 450, 225]  # whole numbers to find the GCF of (any number of them)

greatest_common_divisor = reduce(gcd, numbers)  # applies gcd two at a time from the start

print(f"GCF of {numbers}: {greatest_common_divisor}")

Runs with the standard library only. math.gcd finds the GCF of two whole numbers, and reduce takes care of the "two at a time" part. Replace the list at the top with your own numbers and run it (this example prints 15).

How to write it in LaTeX and other math languages (copy and paste)

Using prime factorization (the basic method)

gcd(a, b) = p₁ × p₂ × ⋯

\gcd(a, b) = p_1 \times p_2 \times \cdots

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>gcd</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo>
    <mo>=</mo>
    <msub><mi>p</mi><mn>1</mn></msub>
    <mo>&#xD7;</mo>
    <msub><mi>p</mi><mn>2</mn></msub>
    <mo>&#xD7;</mo>
    <mo>&#x22EF;</mo>
  </mrow>
</math>

gcd(a, b) = p_1 xx p_2 xx cdots

GCD[a, b]

igcd(a, b);

g = gcd(a, b);

gcd(a, b) = p_1 × p_2 × ⋯

The Euclidean algorithm (the method for large numbers)

gcd(a, b) = gcd(b, a mod b)

\gcd(a, b) = \gcd(b,\ a \bmod b)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>gcd</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo>
    <mo>=</mo>
    <mi>gcd</mi>
    <mo>(</mo><mi>b</mi><mo>,</mo>
    <mi>a</mi><mspace width="0.3em"/><mi>mod</mi><mspace width="0.3em"/><mi>b</mi>
    <mo>)</mo>
  </mrow>
</math>

gcd(a, b) = gcd(b, a mod b)

GCD[a, b] == GCD[b, Mod[a, b]]

igcd(a, b) = igcd(b, a mod b);

gcd(a, b) == gcd(b, mod(a, b))

gcd(a, b) = gcd(b, a mod b)

GCF of 3 or more numbers

gcd(a, b, c) = gcd(gcd(a, b), c)

\gcd(a, b, c) = \gcd(\gcd(a, b),\ c)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>gcd</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo>
    <mo>=</mo>
    <mi>gcd</mi>
    <mo>(</mo>
    <mi>gcd</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo>
    <mo>,</mo><mi>c</mi>
    <mo>)</mo>
  </mrow>
</math>

gcd(a, b, c) = gcd(gcd(a, b), c)

GCD[a, b, c]

igcd(igcd(a, b), c);

g = gcd(gcd(a, b), c);

gcd(a, b, c) = gcd(gcd(a, b), c)

How the GCF is related to the LCM

gcd(a, b) × lcm(a, b) = a × b

\gcd(a, b) \times \mathrm{lcm}(a, b) = a \times b

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>gcd</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo>
    <mo>&#xD7;</mo>
    <mi>lcm</mi>
    <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo>
    <mo>=</mo>
    <mi>a</mi>
    <mo>&#xD7;</mo>
    <mi>b</mi>
  </mrow>
</math>

gcd(a, b) xx lcm(a, b) = a xx b

GCD[a, b]*LCM[a, b] == a*b

igcd(a, b)*ilcm(a, b) = a*b;

gcd(a, b)*lcm(a, b) == a*b

gcd(a, b) × lcm(a, b) = a × b

How to have ChatGPT do the calculation

You are a calculation assistant for whole numbers. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

For the four whole numbers 330, 75, 450 and 225, find each of the following:
1. The greatest common factor (GCF) of the four numbers
2. The prime factorization of each number
3. The prime factors that all four numbers share

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula