Multiple Numbers Greatest Common Factor Calculator

Understanding the Greatest Common Factor (GCF)

What is the Greatest Common Factor?

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given numbers without a remainder. For multiple numbers, it's the largest factor that all the numbers share in common.

Formula and Representation

The GCF of numbers \(a\), \(b\), \(c\), ... is typically represented as:

\[GCF(a,b,c,...)\]

It can be calculated using various methods, including:

1. Listing factors
2. Prime factorization
3. Euclidean algorithm

Euclidean Algorithm for Multiple Numbers

To find the GCF of multiple numbers, we can use the Euclidean algorithm repeatedly:

\[GCF(a,b,c,...) = GCF(GCF(a,b),c,...)\]

Where the GCF of two numbers is calculated as:

\[GCF(a,b) = GCF(b, a \bmod b)\]

Here, \(a \bmod b\) represents the remainder when \(a\) is divided by \(b\).

Example Calculation

Let's find the GCF of 48, 18, and 30:

1. First, find GCF(48, 18):
   • 48 = 18 × 2 + 12
   • 18 = 12 × 1 + 6
   • 12 = 6 × 2 + 0
   • GCF(48, 18) = 6
2. Now, find GCF(6, 30):
   • 30 = 6 × 5 + 0
   • GCF(6, 30) = 6

Therefore, GCF(48, 18, 30) = 6

Visual Representation

This diagram illustrates the GCF of 48, 18, and 30, which is 6. The input numbers are arranged in a circle with their GCF in the center, visually representing how 6 is the largest factor common to all three numbers.