Greatest Common Factor (GCF) Calculator

Find the Largest Common Divisor

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides a set of numbers without leaving a remainder. This calculator uses the efficient Euclidean algorithm to find the GCF for two or more numbers. It's a fundamental concept for simplifying fractions and in number theory.

The Euclidean Algorithm Explained

To find the GCF of two numbers, a and b, the algorithm works as follows:

1. If b is 0, the GCF is a.
2. Otherwise, the GCF is the GCF of b and the remainder of a divided by b (a % b). This process is repeated until the remainder is 0. To find the GCF of more than two numbers, you find the GCF of the first two, then find the GCF of that result and the third number, and so on.

How to Use the Calculator

1. Enter Numbers: Input two or more whole numbers, separated by commas.
2. Calculate GCF: The tool will compute and display the greatest common factor.

Real-World Example

You want to find the GCF of 48 and 18.

• Euclidean Algorithm:
  • GCF(48, 18) → GCF(18, 48 % 18) → GCF(18, 12)
  • GCF(18, 12) → GCF(12, 18 % 12) → GCF(12, 6)
  • GCF(12, 6) → GCF(6, 12 % 6) → GCF(6, 0)
• Since the remainder is 0, the GCF is 6.

If you want the GCF of 48, 18, and 72:

• First, GCF(48, 18) = 6.
• Then, GCF(6, 72) = 6.
• The final GCF is 6.

Frequently Asked Questions (FAQ)

• What is the GCF used for? The most common use is to simplify fractions. To simplify 18/48, you divide both the numerator and the denominator by their GCF (6) to get 3/8.
• What if the numbers have no common factors other than 1? If the GCF of two numbers is 1, they are called 'coprime' or 'relatively prime'.
• Can I use negative numbers? The GCF is typically defined for positive integers. You can enter negative numbers, but the calculator will use their absolute values for the calculation.