GCF Calculator

The Greatest Common Factor (GCF) of two or more numbers is the largest positive integer that divides all the given numbers evenly without leaving a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly.

The most efficient method is the Euclidean algorithm: repeatedly divide the larger number by the smaller number and take the remainder, then continue with the smaller number and the remainder until the remainder is 0. The last non-zero remainder is the GCF. For example, to find GCF(48, 18): 48 ÷ 18 = 2 remainder 12, then 18 ÷ 12 = 1 remainder 6, then 12 ÷ 6 = 2 remainder 0, so GCF = 6.

For multiple numbers, use the associative property: GCF(a, b, c) = GCF(GCF(a, b), c). Find the GCF of the first two numbers, then find the GCF of that result with the third number, and so on.

GCF (Greatest Common Factor) is the largest number that divides all given numbers evenly, while LCM (Least Common Multiple) is the smallest number that is a multiple of all given numbers. They are related by the formula: LCM(a, b) × GCF(a, b) = a × b.

GCF is typically defined for positive integers only. For negative numbers, you would first find the GCF of their absolute values. However, this calculator focuses on positive integers for clarity and practical applications.

GCF is used in simplifying fractions (dividing numerator and denominator by their GCF), scheduling (finding common intervals), engineering (finding optimal gear ratios), and computer science (finding efficient algorithms and data structures).

Prime factorization can be used to find GCF by taking each prime factor to its lowest power across all numbers. For example, for 12 (2²×3) and 18 (2×3²), the GCF is 2×3 = 6, using the lowest power of each prime factor.

GCF is essential for simplifying fractions. To simplify a fraction, divide both the numerator and denominator by their GCF. For example, to simplify 12/18, find GCF(12, 18) = 6, then 12/18 = (12÷6)/(18÷6) = 2/3.

No, the GCF is always less than or equal to the smallest number in the set. The GCF equals the smallest number only when all other numbers are multiples of the smallest number.

To verify, check that the number divides all the original numbers evenly and that no larger positive number has this property. You can also use the relationship: LCM(a, b) × GCF(a, b) = a × b to verify your result.