GCF Calculator

What Is the Greatest Common Factor?

The greatest common factor (GCF) is the largest positive whole number that divides two or more integers without leaving a remainder. For example, the GCF of 12 and 18 is 6 because both numbers are divisible by 6, and no larger positive integer divides them both.

The same value is also called the greatest common divisor (GCD) or highest common factor (HCF). US classrooms usually say GCF; programming libraries usually say GCD. This page computes that shared positive divisor. It does not factor polynomials or algebraic expressions.

How to Use the GCF Calculator

1. Enter two or more integers, separated by commas, spaces, or new lines.
2. Read the GCF as you type, or press Calculate. Invalid tokens are rejected instead of being rounded.
3. Open Factors, Prime factorization, or Euclidean algorithm to inspect the working that matches those methods.

How to Find the Greatest Common Factor

Method 1: Listing Common Factors

List every positive factor of each number, then take the largest value that appears on every list. For 12 and 18, the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The common list is 1, 2, 3, 6, so the GCF is 6. Listing is clear for small homework numbers and slow once the integers grow.

Method 2: Prime Factorization

Write each number as a product of primes, keep only the primes that appear in every factorization, and use the lowest exponent of each. For 360 and 504 the calculator reports 2^3 × 3^2 × 5 and 2^3 × 3^2 × 7. Shared primes at lowest powers give 2^3 × 3^2 = 72. This is the method that also explains GCF versus LCM: GCF uses minimum exponents, LCM uses maximum.

Method 3: Euclidean Algorithm

Replace the pair (a, b) with (b, a mod b) until the remainder is 0. The last non-zero remainder is the GCF. For 1071 and 462:

1. 1,071 = 462 × 2 + 147
2. 462 = 147 × 3 + 21
3. 147 = 21 × 7 + 0

The remainder hits 0 after 21, so GCF(1071, 462) = 21. This is the method the calculator uses for the exact answer, including integers too large to factor quickly.

GCF Formula and Mathematical Method

• gcd(a, b) = gcd(b, a mod b), with gcd(a, 0) = |a| when a ≠ 0.
• GCF(a, b, c) = gcd(gcd(a, b), c). Extra numbers fold in the same way.
• GCF(a, b) = gcd(|a|, |b|). The displayed GCF is a positive integer.
• If a = ∏ p^α and b = ∏ p^β, then gcd(a, b) = ∏ p^min(α,β).
• For nonzero a and b, gcd(a, b) × lcm(a, b) = |a × b|.

gcd(0, 0) is not treated as an ordinary greatest positive factor here. An all-zero list is rejected with that explanation. Decimal input is rejected; this page does not round 12.5 to 12.

GCF Examples With Answers

100 and 250 have GCF 50 because 100 = 2 × 50 and 250 = 5 × 50. 250 ÷ 100 = 2.5, so 100 is not a factor of 250.

GCF of Two, Three, or More Numbers

For two numbers, one Euclidean chain is enough: GCF(24, 36) = 12. For three numbers, fold the result into the next value:

GCF(24, 36) = 12

GCF(12, 60) = 12

Therefore GCF(24, 36, 60) = 12

The same chain extends to the 12-integer limit on this page. Coprime sets such as 8 and 15 still have a GCF: 1.

Where Is GCF Used?

• Simplifying fractions. Divide numerator and denominator by GCF(a, b). 24/36 becomes 2/3. After that reduction, no integer greater than 1 divides both parts. Use the Fraction Calculator when you also need to add or subtract.
• Reducing ratios. The same 24 : 36 pair reduces to 2 : 3. Continue in the Ratio Calculator.
• Identical groups. 24 pencils and 36 erasers can fill 12 identical bags with 2 pencils and 3 erasers each. 12 is the number of bags, not the number of items in a bag.
• Equal segments. Boards of 84 in and 126 in cut into 42 in pieces with nothing left over.
• Square tiles on an integer grid. A 48 by 72 rectangle tiles with 24 × 24 squares.

GCF vs. LCM

GCF(12, 18) = 6. The least common multiple of the same pair is 36. Factors divide a number; multiples are produced by multiplying it. Use GCF to simplify. Use the LCM Calculator when you need a common denominator or a first shared multiple. For nonzero a and b, GCF × LCM = |a × b|.

Common Mistakes When Finding GCF

• Listing multiples instead of factors, then confusing the answer with an LCM.
• Stopping at a common factor such as 2 when the greatest shared factor is larger.
• Treating GCF = 1 as a failure. Relatively prime integers are allowed.
• Using maximum prime exponents (the LCM rule) instead of minimum exponents.
• Displaying GCF(0, 0) = 0 as if it were an ordinary greatest positive factor. This page refuses that list.
• Keeping a negative GCF. The value is gcd(|a|, |b|); -12, 18 still gives 6.
• Rounding a decimal so the tool can continue. 12.5 is rejected.
• Using GCF when adding unlike fractions. That task needs an LCM for the denominators.

Related Math Calculators

• LCM Calculator — Least common multiple for shared multiples and denominators.
• Fraction Calculator — Simplify or operate on fractions after you have the GCF.
• Prime Number Calculator — Test primes and inspect factors used in the prime method.
• Ratio Calculator — Reduce a ratio by the same shared divisor.

Prime factorization in the result panel pairs with the Prime Number Calculator.

Frequently Asked Questions