GCF / GCD Calculator
Greatest Common Factor & Divisor
Enter two or three whole numbers to find their Greatest Common Factor.
GCF / GCD
1
Greatest Common Factor
Euclidean Algorithm
Prime Factorizations
Relationships
GCF
LCM
GCF × LCM
Ad · 300x250
Greatest Common Factor Methods & Uses
The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest number that divides evenly into all given numbers with no remainder. GCF is essential for simplifying fractions and solving real-world division problems.
The Euclidean algorithm is the most efficient method: divide the larger number by the smaller, take the remainder, and repeat until the remainder is zero. The last non-zero remainder is the GCF.
Euclidean Algorithm
GCF(48, 18): 48 = 2×18 + 12. GCF(18,12): 18 = 1×12 + 6. GCF(12,6): 12 = 2×6 + 0. Answer: GCF = 6. Repeat dividing and taking remainders until 0.
Prime Factorization Method
48 = 2³×3. 18 = 2×3². GCF = product of common prime factors with lowest exponents = 2¹×3¹ = 6. Clear but slow for large numbers.
Simplifying Fractions with GCF
18/48: GCF(18,48) = 6. 18/6 = 3, 48/6 = 8. Simplified: 3/8. Always divide both numerator and denominator by their GCF.
GCF vs LCM Relationship
GCF(a,b) × LCM(a,b) = a × b. For 48 and 18: GCF = 6, LCM = (48×18)/6 = 144. Knowing GCF lets you calculate LCM instantly.
Ad · 728x90
GCF Questions
What is the Greatest Common Factor (GCF)?+
The GCF of two or more numbers is the largest number that divides evenly into all of them with no remainder. GCF(12, 18) = 6, because 6 divides both 12 and 18, and no number larger than 6 does. GCF is also called Greatest Common Divisor (GCD) or Highest Common Factor (HCF). These terms all refer to the same concept.
How do I find the GCF using the Euclidean algorithm?+
Divide the larger number by the smaller and find the remainder. Replace the larger with the smaller, and the smaller with the remainder. Repeat until the remainder is 0. The last non-zero number is the GCF. Example GCF(56, 98): 98 = 1×56 + 42. Then GCF(56, 42): 56 = 1×42 + 14. Then GCF(42, 14): 42 = 3×14 + 0. GCF = 14. This algorithm is efficient even for very large numbers.
How do I find GCF using prime factorization?+
Factor each number into its prime factors. Identify the primes that appear in all factorizations. Take the lowest power of each common prime. Multiply these together. Example: GCF(60, 90). 60 = 2² × 3 × 5. 90 = 2 × 3² × 5. Common primes: 2 (power 1), 3 (power 1), 5 (power 1). GCF = 2 × 3 × 5 = 30.
What is GCF used for in real life?+
Simplifying fractions (divide numerator and denominator by GCF). Dividing into equal groups without leftovers: 24 apples and 36 oranges can be divided into GCF(24,36) = 12 equal groups, each with 2 apples and 3 oranges. Solving tile and grid problems: the largest square tile that fits a 12 ft by 18 ft room without cutting has side GCF(12,18) = 6 ft. Reducing ratios to simplest form.
What is the GCF of two prime numbers?+
Always 1. Prime numbers have no factors other than 1 and themselves. Two different primes share no common factors, so GCF(7, 11) = 1, GCF(13, 17) = 1. Numbers whose GCF is 1 are called coprime or relatively prime. They do not need to be prime themselves: GCF(8, 9) = 1, even though neither 8 nor 9 is prime.
How do I find the GCF of three numbers?+
Find GCF of the first two numbers, then find GCF of that result and the third number. Example: GCF(12, 18, 24). GCF(12, 18) = 6. GCF(6, 24) = 6. Answer: 6. This works because GCF is associative: GCF(a, b, c) = GCF(GCF(a, b), c). You can extend this to as many numbers as needed.
What is the difference between GCF and LCM?+
GCF is the LARGEST number that divides all given numbers. LCM is the SMALLEST number that all given numbers divide into. GCF(4, 6) = 2, LCM(4, 6) = 12. They are related: GCF × LCM = product of the two numbers (for two numbers). GCF is used to simplify fractions; LCM is used to find common denominators for adding fractions. GCF ≤ min(a,b) and LCM ≥ max(a,b).
What is GCF(0, n) for any number n?+
GCF(0, n) = n for any positive integer n. This is because every positive integer divides 0 (0 = n × 0), so the greatest divisor of both 0 and n is n itself. The Euclidean algorithm confirms this: GCF(0, n) divides until remainder = 0 immediately, leaving n as the GCF. GCF(0, 0) is typically defined as 0 by convention.
How do I use GCF to simplify a fraction?+
Divide both the numerator and denominator by their GCF. Example: simplify 36/48. GCF(36, 48) = 12. 36/12 = 3, 48/12 = 4. Simplified fraction: 3/4. This works because dividing both parts of a fraction by the same number does not change its value. The result is in lowest terms when GCF(numerator, denominator) = 1.
Is GCF the same as GCD and HCF?+
Yes, all three terms refer to the exact same mathematical concept. GCF (Greatest Common Factor) is most common in US elementary and middle school education. GCD (Greatest Common Divisor) is preferred in higher mathematics, computer science, and number theory. HCF (Highest Common Factor) is the standard term in UK, India, and many Commonwealth countries. They all mean: the largest positive integer that divides all given numbers without remainder.