What is the greatest common factor (GCF)?

The greatest common factor (GCF) of a set of integers is the largest positive integer that divides each of them without remainder. It is also called the greatest common divisor (GCD). For example, the GCF of 18 and 24 is 6 because the common factors are 1, 2, 3 and 6, and 6 is the largest.

Is GCF the same as GCD?

Yes. In arithmetic and number theory, greatest common factor (GCF) and greatest common divisor (GCD) are two names for the same concept. GCF is more common in school arithmetic, while GCD is more common in proofs and advanced texts.

How do you compute the GCF using the Euclidean algorithm?

To compute the GCF of two integers a and b (with a ≥ b > 0), repeatedly divide and take remainders: compute a = q1·b + r1, then replace a with b and b with r1, and repeat until the remainder is 0. The last non-zero remainder is the GCF. This process is much faster than listing all factors for large numbers.

Can I find the GCF of more than two numbers?

Yes. The GCF of several numbers can be found by applying the Euclidean algorithm pairwise, for example GCF(a, b, c) = GCF(GCF(a, b), c). The calculator does this automatically when you enter a list of numbers.

Does this calculator work with negative numbers or zero?

The GCF is defined for integers, and by convention we take it to be non-negative. The calculator uses absolute values, so changing the sign of a number does not change the GCF. At least one of the numbers must be non-zero; if all numbers are 0 the GCF is undefined.

GCF Calculator – Greatest Common Factor of 2 or More Numbers

Definition of greatest common factor (GCF)

For integers \(a_1, a_2, \dots, a_k\) (not all zero), the greatest common factor \(\gcd(a_1, a_2, \dots, a_k)\) is the largest positive integer that divides each \(a_i\) without remainder:

\[ \gcd(a_1, a_2, \dots, a_k) = \max\{ d \in \mathbb{N} \,:\, d \mid a_i \text{ for all } i \}. \]

In elementary arithmetic this is called the greatest common factor (GCF), while in number theory it is more often called the greatest common divisor (GCD).

The Euclidean algorithm

The Euclidean algorithm is the standard efficient method to compute the GCF of two integers. For \(a, b > 0\), with \(a \ge b\):

Compute the remainder: \(a = q_1 b + r_1\), with \(0 \le r_1 < b\).
If \(r_1 = 0\), then \(\gcd(a, b) = b\).
Otherwise, replace \((a, b)\) by \((b, r_1)\) and repeat until the remainder is 0.

The last non-zero remainder is the GCF. This works because \(\gcd(a, b) = \gcd(b, r_1) = \gcd(r_1, r_2) = \dots\).

Prime factorization method

Another way to compute the GCF is via prime factorization:

If \[ a = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_m^{\alpha_m}, \quad b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_m^{\beta_m} \] where \(p_i\) are the primes appearing in at least one factorization, then \[ \gcd(a, b) = p_1^{\min(\alpha_1, \beta_1)} p_2^{\min(\alpha_2, \beta_2)} \cdots p_m^{\min(\alpha_m, \beta_m)}. \]

This method is very transparent but less efficient for large numbers than the Euclidean algorithm. Our calculator uses Euclid’s method for speed and prime factorization for explanation.

Why GCF matters: applications

Simplifying fractions: to reduce \(\frac{a}{b}\) to lowest terms, divide both numerator and denominator by \(\gcd(a, b)\).
Ratios and proportions: the simplest integer ratio representing quantities \(a_1, a_2, \dots, a_k\) is obtained by dividing each by the GCF of the list.
Number theory: coprime integers (GCF = 1) are fundamental in modular arithmetic, Diophantine equations and cryptography.
LCM and GCF: the least common multiple and the GCF are linked by \(\gcd(a, b)\, \mathrm{lcm}(a, b) = |ab|\) for non-zero integers a, b.

FAQ: using this GCF calculator

How many numbers can I enter?

Practically, you can enter anywhere from 2 up to a few dozen integers. The algorithm scales well, but extremely long lists may be slower. The calculator automatically applies the Euclidean algorithm pairwise: \(\gcd(a_1, a_2, \dots, a_k) = \gcd(\dots \gcd(a_1, a_2), a_3), \dots, a_k)\).

Does the order of the numbers matter?

No. The GCF is symmetric: \(\gcd(a_1, a_2, \dots, a_k)\) is the same regardless of the order. However, for human-readable Euclidean steps we focus on the first two numbers you entered.

Can the GCF ever be larger than the smallest number?

No. The GCF is always less than or equal to the smallest non-zero absolute value among the inputs. If one of the numbers is 0, \(\gcd(a, 0) = |a|\) by convention (provided a ≠ 0).

Is this calculator suitable for exam or homework use?

The tool is designed to be transparent: it shows steps, factorizations and explanations so you can learn and verify your own work. For exams, always follow your teacher’s rules on calculator use and be prepared to explain the steps, not just the final GCF.

GCF Calculator (Greatest Common Factor)

1. Enter the integers

2. Result summary

3. Small GCF table (examples)