Find the GCF and LCM of 2-4 numbers using the Euclidean algorithm and prime factorization. View step-by-step solutions, Bézout coefficients, and factor comparison visuals.
The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It is a foundational concept in number theory, algebra, and applied mathematics, used for simplifying fractions, factoring polynomials, solving Diophantine equations, and designing algorithms in computer science.
Two classical methods exist for computing the GCF. The first is prime factorization: decompose each number into its prime factors, then multiply together the lowest powers of all common primes. For example, 12 = 2² × 3 and 18 = 2 × 3², so GCF(12, 18) = 2¹ × 3¹ = 6. The second is the Euclidean algorithm, one of the oldest known algorithms, which repeatedly divides the larger number by the smaller and replaces it with the remainder until the remainder is zero. The last non-zero remainder is the GCF.
This calculator supports 2 to 4 numbers and displays both methods side by side. It also computes the least common multiple (LCM), the simplified ratio of all inputs, and—for two numbers—the Bézout coefficients x and y such that ax + by = GCF. The prime factorization table lets you visually compare which primes are shared and at what powers. The factor proportion bars show how much of each number the GCF represents, giving an intuitive sense of the relationship between the inputs.
Whether you are simplifying a fraction for homework, computing gear ratios in engineering, or studying modular arithmetic, this tool provides a complete, step-by-step breakdown.
Greatest Common Factor (GCF) Calculator helps you solve greatest common factor (gcf) problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Numbers (comma-separated, 2-4) once and immediately inspect GCF (Greatest Common Factor), LCM (Least Common Multiple), Product of Numbers to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Euclidean: GCD(a, b) = GCD(b, a mod b) until remainder is 0. LCM(a, b) = |a × b| / GCD(a, b). Bézout: ax + by = GCD(a, b).
Result: GCF (Greatest Common Factor) shown by the calculator
Using the preset "12, 18", the calculator evaluates the greatest common factor (gcf) setup, applies the selected algebra rules, and reports GCF (Greatest Common Factor) with supporting checks so you can verify each transformation.
This calculator takes Numbers (comma-separated, 2-4) and applies the relevant greatest common factor (gcf) relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use GCF (Greatest Common Factor), LCM (Least Common Multiple), Product of Numbers, GCF × LCM to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
They are the same concept. GCF (Greatest Common Factor) and GCD (Greatest Common Divisor) are interchangeable terms. GCF is more common in U.S. school math, while GCD is standard in higher mathematics and computer science.
Divide the larger number by the smaller, note the remainder, then replace the larger number with the smaller and the smaller with the remainder. Repeat until the remainder is zero. The last non-zero remainder is the GCF.
Bézout's identity states that for any integers a and b, there exist integers x and y such that ax + by = GCD(a, b). These coefficients are useful in solving linear Diophantine equations and in modular inverse calculations.
No. The GCF is always less than or equal to the smallest of the input numbers. It equals the smallest input only if the smallest number divides all other inputs.
Compute the GCF of the first two numbers, then compute the GCF of that result with the third number, and so on. GCF is associative: GCF(a, b, c) = GCF(GCF(a, b), c).
For two numbers a and b, GCF(a,b) × LCM(a,b) = |a × b|. This identity does not extend directly to three or more numbers.