Lesson 20: Greatest Common Factor (GCF)

New Concept

The greatest common factor (GCF) is the largest whole number that is a factor of two or more given numbers.

What’s next

This is just the foundation. Next, you'll tackle worked examples to see the process in action and then test your skills on various problem sets.

Property

The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder.

Examples

• For 12 and 18, the factors of 12 are $1, 2, 3, 4, 6, 12$ and the factors of 18 are $1, 2, 3, 6, 9, 18$. The GCF is $6$.
• For 6, 9, and 15, the factors of 6 are $1, 2, 3, 6$; factors of 9 are $1, 3, 9$; factors of 15 are $1, 3, 5, 15$. The GCF is $3$.
• For 10 and 15, the factors of 10 are $1, 2, 5, 10$ and the factors of 15 are $1, 3, 5, 15$. The GCF is $5$.

Explanation

Imagine two numbers showing off their factor collections. The GCF is the biggest, coolest factor they BOTH have. To find it, you list all the factors for each number, find all the factors they have in common, and then pick the largest one from that shared group. It's the king of all common factors!

Property

The greatest common factor (GCF) of any two different prime numbers is always 1.

Examples

• The GCF of 7 (factors: 1, 7) and 13 (factors: 1, 13) is $1$.
• The GCF of 3 (factors: 1, 3) and 19 (factors: 1, 19) is $1$.
• The GCF of 23 (factors: 1, 23) and 5 (factors: 1, 5) is $1$.

Explanation

Prime numbers are famously picky, with only 1 and themselves as factors. So, when you compare two different primes, like 5 and 11, the only factor they have in common is always the number 1. This makes finding their GCF super easy—it’s the one and only common factor they could possibly share!