GCF Calculator
Find the Greatest Common Factor (GCF/GCD) of up to 10 numbers using the Euclidean algorithm, with step-by-step explanation.
About GCF / GCD
The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), is the largest positive integer that divides each of the given numbers without a remainder. It is used to simplify fractions, factor polynomials, and solve problems involving ratios and proportions.
The Euclidean Algorithm
The Euclidean algorithm is an efficient method for computing the GCD. It is based on the principle that GCD(a, b) = GCD(b, a mod b). The algorithm repeatedly replaces the larger number with the remainder of dividing the two numbers until the remainder is zero. The last non-zero remainder is the GCD.
GCF and LCM Relationship
For any two integers a and b, the product of their GCF and LCM equals the product of the two numbers: GCF(a,b) × LCM(a,b) = a × b. This relationship is extremely useful — if you know either the GCF or LCM, you can compute the other instantly. For multiple numbers, apply the relationship iteratively.
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For a related calculation, use Fraction Calculator to add, subtract, multiply, and divide fractions with step-by-step solutions. Alternatively, use LCM Calculator to calculate the Least Common Multiple of up to 10 numbers with step-by-step prime factorization.