GCF Calculator — Greatest Common Factor
Last updated: February 12, 2026
The Greatest Common Factor (GCF) is the largest positive integer that divides two or more numbers without a remainder. To find it, enter your numbers below — this free calculator shows the answer instantly with step-by-step solutions using three proven methods: listing factors, prime factorization, and the Euclidean algorithm.
GCF Calculator
💡 Use the GCF to Simplify Fractions
Divide both the numerator and denominator by the GCF to reduce any fraction to its simplest form. Try our Simplify Fractions Calculator or learn How to Simplify Fractions.
What Is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. The GCF is also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF). Every pair of positive integers shares at least one common factor: the number 1.
Key definition: GCF(a, b) is the largest number d such that a ÷ d and b ÷ d are both whole numbers.
Why Is the GCF Important for Fractions?
The GCF plays a central role in working with fractions. When you simplify a fraction, you divide the numerator and denominator by their GCF. For instance, the fraction 12/18 simplifies to 2/3 because GCF(12, 18) = 6, and dividing both parts by 6 gives the reduced form.
How Are GCF and LCM Related?
The GCF is closely related to the Least Common Multiple (LCM). For any two positive integers a and b, the relationship is: GCF(a, b) × LCM(a, b) = a × b. So if you know one, you can quickly find the other.
How Do You Find the GCF?
You can find the Greatest Common Factor using three methods: listing all factors, prime factorization, or the Euclidean algorithm. The listing method works best for small numbers, prime factorization is the most commonly taught approach, and the Euclidean algorithm is the fastest for large numbers.
How Do You Find the GCF by Listing Factors?
To find the GCF by listing factors, write out every factor of each number, then identify the largest factor that appears in every list. This method is intuitive and works well for small numbers under 50, but becomes tedious for larger values.
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
GCF = 12 (the largest common factor)
How Do You Find the GCF Using Prime Factorization?
To find the GCF using prime factorization, break each number into its prime factors, identify the primes they share, take the lowest power of each shared prime, and multiply them together. This is the most commonly taught method in schools and scales well to larger numbers.
Find GCF(48, 180):
48 = 2⁴ × 3¹
180 = 2² × 3² × 5¹
Common primes at lowest powers: 2² × 3¹ = 4 × 3
GCF = 12
How Does the Euclidean Algorithm Find the GCF?
The Euclidean algorithm finds the GCF by repeatedly dividing the larger number by the smaller, then replacing the larger with the remainder, until the remainder reaches 0. The last non-zero remainder is the GCF. Developed around 300 BC, this is the most efficient method and is widely used in computer science and cryptography.
Find GCF(48, 18):
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0
GCF = 6 (the last non-zero remainder)
Which GCF Method Should You Use?
| Method | Best For | Difficulty |
|---|---|---|
| Listing Factors | Small numbers (under 50) | Easy |
| Prime Factorization | Medium numbers, learning | Medium |
| Euclidean Algorithm | Large numbers, speed | Easy once learned |
Frequently Asked Questions About the GCF
The Greatest Common Factor (GCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6 because 6 is the biggest whole number that goes into both 12 and 18 evenly. The GCF is also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).
The GCF (Greatest Common Factor) is the largest number that divides all given numbers evenly, while the LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly. For 12 and 18, the GCF is 6 and the LCM is 36. They are related by the formula: GCF × LCM = product of the two numbers. Learn more with our LCD/LCM Calculator.
To find the GCF using prime factorization, follow four steps. First, find the prime factorization of each number. Second, identify the prime factors that appear in every number. Third, take the lowest exponent for each shared prime. Fourth, multiply those values together. For example, 24 = 2³ × 3 and 36 = 2² × 3², so GCF = 2² × 3¹ = 12.
The GCF of 1 and any other number is always 1. Since the only positive factor of 1 is 1 itself, the only factor that 1 can share with any other number is 1. When two numbers have a GCF of 1, they are called coprime or relatively prime.
To simplify a fraction using the GCF, divide both the numerator and the denominator by their Greatest Common Factor. For example, to simplify 12/18, first find GCF(12, 18) = 6. Then divide: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. The simplified fraction is 2/3. Try our Simplify Fractions Calculator to do this automatically.
Yes, you can find the GCF of three or more numbers by chaining the process. Find the GCF of the first two numbers, then find the GCF of that result with the third number, and continue for each additional number. For example, GCF(12, 18, 24) = GCF(GCF(12, 18), 24) = GCF(6, 24) = 6. This calculator supports up to 6 numbers at once.
The Euclidean algorithm finds the GCF by repeatedly dividing the larger number by the smaller and replacing the larger number with the remainder. When the remainder reaches zero, the last non-zero remainder is the GCF. For example, for GCF(48, 18): 48 ÷ 18 = 2 R12, then 18 ÷ 12 = 1 R6, then 12 ÷ 6 = 2 R0. The GCF is 6.
When the GCF of two numbers is 1, the numbers are called coprime or relatively prime. This means they share no common factors other than 1. For example, 8 and 15 are coprime because GCF(8, 15) = 1. A fraction whose numerator and denominator are coprime — like 8/15 — is already in its simplest form and cannot be reduced further.
The Euclidean algorithm is the fastest way to find the GCF of large numbers. Unlike listing factors or prime factorization, it requires only a few division steps regardless of how large the numbers are. For example, finding GCF(462, 1071) takes just three steps: 1071 ÷ 462 = 2 R147, 462 ÷ 147 = 3 R21, 147 ÷ 21 = 7 R0. The GCF is 21.
Yes, the GCF of two even numbers is always even. Since both even numbers are divisible by 2, they always share at least 2 as a common factor. The GCF could be 2 itself or a larger even number. For example, GCF(8, 12) = 4 and GCF(6, 10) = 2. Both results are even numbers, which holds true for any pair of even integers.