Find the Greatest Common Factor (GCF) of two or more numbers instantly with step-by-step prime factorization.
Enter two or more numbers separated by commas to find their Greatest Common Factor.
A GCF calculator (Greatest Common Factor calculator) is a mathematical tool that finds the largest positive integer that divides two or more numbers without leaving a remainder. The GCF is also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF). This concept is fundamental in number theory and is widely used in simplifying fractions, solving ratio problems, and various applications in algebra and arithmetic.
For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly. Understanding GCF helps students build a strong foundation in mathematics and is essential for standardized tests like the SAT, GRE, and GMAT. According to Khan Academy, mastering GCF is a key skill for 6th-grade mathematics and beyond.
The prime factorization method breaks each number into its prime factors, then identifies the common factors. This is the most intuitive approach taught in schools and recommended by Math is Fun.
Example: Find GCF of 48 and 36
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹
36 = 2 × 2 × 3 × 3 = 2² × 3²
Common factors: 2² × 3¹ = 4 × 3 = 12
The Euclidean Algorithm is an efficient method that uses repeated division. It was first described by the Greek mathematician Euclid around 300 BCE and remains one of the oldest algorithms still in use today, as noted by Wikipedia.
Example: Find GCF of 48 and 18
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0
GCF = 6
For smaller numbers, simply list all factors and find the largest one in common. This method works well for quick mental calculations and is perfect when working with numbers under 100.
Example: Find GCF of 20 and 30
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Common: 1, 2, 5, 10 → GCF = 10
The GCF and LCM (Least Common Multiple) are related but opposite concepts. The GCF finds the largest shared divisor, while the LCM finds the smallest shared multiple. There's a useful relationship between them:
For instance, GCF(12, 18) = 6 and LCM(12, 18) = 36. Verification: 6 × 36 = 216 = 12 × 18 ✓. This property is extensively covered in the Purplemath GCF guide.
The GCF of 1 and any number is always 1, since 1 is the only positive integer that divides 1.
Only if they are the same prime number. Two different prime numbers always have a GCF of 1 because prime numbers have no common factors other than 1. Such numbers are called coprime or relatively prime.
To simplify a fraction, divide both the numerator and denominator by their GCF. For example, to simplify 24/36: GCF(24,36) = 12, so 24/36 = (24÷12)/(36÷12) = 2/3.
They are all the same thing! GCF (Greatest Common Factor), GCD (Greatest Common Divisor), and HCF (Highest Common Factor) all refer to the largest number that divides two or more numbers without a remainder. Different countries and textbooks use different terms.