Greatest Common Factor Calculator GCF
Greatest Common Factor Calculator
Calculate GCF
Enter at least 2 positive integers separated by commas. Results will calculate automatically.
Instructions: Enter positive integers separated by commas (e.g., 12, 18, 24). The calculator will automatically find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) with detailed steps.
The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), is the largest number that divides evenly into two or more numbers. It is essential for simplifying fractions, factoring algebraic expressions, and solving ratio problems. This calculator finds the GCF of multiple numbers with step-by-step explanation.
Methods to Find GCF
There are three main methods. The listing method lists all factors of each number and finds the largest common factor. The prime factorization method finds prime factorizations and multiplies common prime factors. The Euclidean algorithm is fastest for large numbers.
Euclidean Algorithm for GCF(48, 18): 48 = 2 × 18 + 12 18 = 1 × 12 + 6 12 = 2 × 6 + 0 GCF = 6 (last non-zero remainder) Prime factorization method: 48 = 2⁴ × 3 18 = 2 × 3² GCF = 2¹ × 3¹ = 6
Using GCF to Simplify Fractions
Divide both numerator and denominator by their GCF to get the simplest form of a fraction.
Simplify 48/18: GCF(48,18) = 6 48/6 = 8, 18/6 = 3 Simplified: 8/3
Frequently Asked Questions
What is the relationship between GCF and LCM?⌄
GCF × LCM = Product of the two numbers. For 12 and 18: GCF = 6, LCM = 36. Check: 6 × 36 = 216 = 12 × 18. This relationship allows you to find LCM if you know GCF: LCM = (a × b) / GCF.
When is the GCF equal to 1?⌄
The GCF is 1 when the two numbers share no common prime factors. Such numbers are called coprime or relatively prime. Examples: GCF(8,15)=1 (8=2³, 15=3×5, no shared primes). Fractions whose numerator and denominator are coprime are already in simplest form.
How do I find the GCF of three or more numbers?⌄
Apply the GCF operation repeatedly. GCF(a,b,c) = GCF(GCF(a,b),c). Find GCF of the first two numbers, then find GCF of that result with the third number, and so on. Example: GCF(12,18,24) = GCF(GCF(12,18),24) = GCF(6,24) = 6.
What is the GCF used for in real-world problems?⌄
Dividing things into equal groups (GCF gives the largest equal group size), tiling problems (GCF gives the largest square tile size for a rectangle), simplifying ratios and fractions, and distributing items equally without leftovers. Example: 12 apples and 18 oranges distributed into identical bags: GCF(12,18)=6 bags, each with 2 apples and 3 oranges.