How to Use the GCF Calculator
Enter at least two integers, separated by commas or spaces. The calculator finds the greatest positive common divisor of the entered values under its stated input rules.
Review the GCF first. Open the available solution method to see how the result was obtained. If additional results such as the LCM or simplified ratio are displayed, treat them as related calculations.
Enter Integers
Type two or more valid whole integers into the input field, separated by commas, spaces, or tabs.
Read Primary GCF
Instantly view the prominent GCF result, alongside the concise educational explanation line.
Explore Derivations
Switch between Prime Factorization, Euclidean Algorithm, Division Ladder, Factors, or Bézout's identity.
What Is the Greatest Common Factor (GCF)?
The greatest common factor is the largest positive integer that divides every number in a set without leaving a remainder. It is also called the greatest common divisor (GCD) or the highest common factor (HCF).
For example, the positive factors shared by 12 and 18 are 1, 2, 3 and 6. The largest of these is 6, so:
A common factor divides every number in the set. The greatest common factor is the largest such factor. In ordinary integer calculations, GCF, GCD, and HCF refer to the same mathematical quantity.
Worked Example: Finding the GCF
Consider the numbers 36, 54 and 90.
Their prime factorizations are:
36 = 2² × 3²
54 = 2 × 3³
90 = 2 × 3² × 5
The primes shared by all three numbers are 2 and 3. For each shared prime, select the smallest exponent appearing across the factorizations:
- Prime 2: minimum exponent is 1.
- Prime 3: minimum exponent is 2.
Multiply the shared prime powers:
The number 18 divides all three values exactly, and no larger positive integer divides them all.
Methods to Find the GCF
A GCF can be found in several ways. The calculator supports multiple exact mathematical methods to inspect the derivation:
Listing Factors
For small integers, list the positive factors of each number. Identify the factors common to all the numbers, then choose the largest.
For example, the common factors of 12 and 18 are 1, 2, 3 and 6, so their GCF is 6.
Prime Factorization
Write every input as a product of prime powers. Keep only primes shared by all the inputs, using the smallest exponent for each shared prime. Multiply those prime powers to obtain the GCF.
Euclidean Algorithm
The Euclidean algorithm repeatedly replaces a pair of integers with the smaller integer and the remainder from division. It uses the identity:
Repeat until the remainder is zero. The last non-zero divisor is the GCF. For example:
180 = 48 × 3 + 36
48 = 36 × 1 + 12
36 = 12 × 3 + 0
The last non-zero remainder is 12, so GCF(180, 48) = 12.
Division Grid / Ladder Method
Commonly taught in school mathematics, this method places the numbers side-by-side on a horizontal row and divides them simultaneously by common prime divisors. When the remaining quotients share no factor greater than 1, multiplying all extracted divisors gives the GCF.
GCF of Three or More Numbers
The same process applies to more than two integers. The GCF of multiple values can be found by applying the GCF operation successively:
For example:
The GCF does not depend on the order of the input numbers. Reordering the values produces the exact same result.
GCF vs. LCM
The greatest common factor (GCF) and least common multiple (LCM) answer different questions:
- GCF: The largest positive integer that divides every input exactly.
- LCM: The smallest positive common multiple of the input integers, under the usual positive-integer convention.
For two positive integers a and b:
For 48 and 180, the GCF is 12 and the LCM is 720. Their product is 8,640, which equals 48 × 180.
Use the GCF when finding the largest common divisor or reducing a ratio. Use the LCM Calculator when looking for the smallest shared multiple.
Uses of the Greatest Common Factor
The GCF is useful when simplifying fractions, reducing ratios and dividing quantities into equal groups without leftovers:
Simplifying fractions
Divide the numerator and denominator by their GCF. For example:
48/180 = (48 ÷ 12)/(180 ÷ 12) = 4/15
For complete fraction operations including addition and multiplication, see our Fraction Calculator.
Simplifying ratios
Divide every term by the GCF of all the ratio terms. For example:
36 : 54 : 90 = 2 : 3 : 5
Equal grouping
If 36 pens, 54 notebooks and 90 folders are to be distributed into the largest possible number of identical kits with no leftovers, the GCF is 18 kits. Each kit receives 2 pens, 3 notebooks and 5 folders.
Input Rules and Special Cases
Enter integer values rather than decimals or algebraic expressions. The calculator supports whole integers within JavaScript's safe integer limit (-9,007,199,254,740,991 to 9,007,199,254,740,991).
For signed integers, the conventional non-negative GCF is calculated from the absolute values. For example:
For a non-zero integer a:
The result when all inputs are zero follows the mathematical convention that the GCF of all-zero inputs is undefined (since every non-zero integer divides 0, leaving no greatest divisor). The calculator displays an explanatory validation message in this case.
Invalid inputs must be corrected before the calculator displays a current result.
Frequently Asked Questions
What is the GCF?
The GCF is the largest positive integer that divides every number in a set without a remainder.
Is GCF the same as GCD and HCF?
For ordinary integer calculations, yes. GCF means greatest common factor, GCD means greatest common divisor, and HCF means highest common factor.
How do I find the GCF of two numbers?
You can list their factors, use prime factorization, or apply the Euclidean algorithm. Enter the two integers into the calculator to obtain the result and available working.
How do I find the GCF of three or more numbers?
Enter all supported integers in the same input field. The GCF of the complete set is the greatest positive integer that divides every input exactly.
What is the GCF of 36 and 180?
The GCF is 36 because 180 is divisible by 36, and no common factor can be larger than the smaller positive input.
What does it mean when the GCF is 1?
It means the inputs have no common divisor greater than 1. For a set of integers, this is commonly described as being collectively relatively prime.
Can the GCF be found for negative numbers or zero?
For signed integers, the conventional non-negative GCF uses the absolute values. A non-zero integer and zero have a GCF equal to the absolute value of the non-zero integer. The all-zero case follows the calculator's documented convention.
Can this calculator find the GCF of monomials?
This calculator accepts integers, not algebraic expressions containing variables. Finding the GCF of monomials also requires comparing variable factors and their exponents, so it requires a symbolic algebra calculation rather than the integer-only calculation provided here.
Related Math Calculators
LCM Calculator
Find the least common multiple of two or more integers with step-by-step prime decompositions.
Fraction Calculator
Add, subtract, multiply, divide, and reduce fractions to their simplest irreducible form using GCF.
Factor Calculator
Enumerate all positive and negative integer divisors, factor pairs, and prime factorizations.