Greatest Common Factor Calculator

What Is the Greatest Common Factor (GCF)?

The Greatest Common Factor (GCF)—also referred to as the Greatest Common Divisor (GCD)—is the largest positive number that can evenly divide two or more non-zero integers. In simpler terms, it’s the biggest whole number that divides each of the given numbers without leaving a remainder. For instance, the GCF of 32 and 256 is 32, written as GCF(32, 256) = 32.

How to Find the GCF

There are several techniques to calculate the greatest common factor. Two of the most common methods are Prime Factorization and the Euclidean Algorithm. Let’s explore both:

1. Prime Factorization Method

This method involves breaking each number down into its prime factors, identifying the factors they have in common, and multiplying those common factors to find the GCF.

Example:

The common prime factors are 2 × 2 × 2, so:

GCF(16, 88, 104) = 8

This method works well for smaller numbers but can become time-consuming for larger ones.

2. Euclidean Algorithm (Efficient for Large Numbers)

The Euclidean algorithm is a faster and more efficient method to compute the GCF, especially with large integers. It is based on the principle that the GCF of two numbers also divides their difference.

Steps:

• If both numbers are the same:
  GCF(a, a) = a

• If a > b:
  GCF(a, b) = GCF(a – b, b)

• If b > a:
  GCF(a, b) = GCF(a, b – a)

This process continues until the remainder is 0. The GCF is the last non-zero remainder.

Example:

Find GCF(268442, 178296)

1. 268442 – 178296 = 90146

2. 178296 – 90146 = 88150

3. 90146 – 88150 = 1996

4. 88150 ÷ 1996 = 44 remainder 326

5. 1996 ÷ 326 = 6 remainder 40

6. 326 ÷ 40 = 8 remainder 6

7. 40 ÷ 6 = 6 remainder 4

8. 6 ÷ 4 = 1 remainder 2

9. 4 ÷ 2 = 2 remainder 0

The last non-zero remainder is 2, so:
GCF(268442, 178296) = 2

Handling More Than Two Numbers

If you’re finding the GCF for three or more numbers, simply repeat the process. For example:

GCF(268442, 178296, 66888)
First, find GCF(268442, 178296) = 2, then compute GCF(2, 66888) = 2

So the final answer is:
GCF(268442, 178296, 66888) = 2