Greatest Common Factor (GCF) - Definition, Examples, Quiz, FAQ, Trivia

What is the Greatest Common Factor?

The Greatest Common Factor (GCF) is the largest number that divides exactly into two or more numbers. It's also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

Why is GCF important? We use it to simplify fractions, factor polynomials, and solve real-world problems involving sharing or grouping items equally.

Here's what you need to know:

• Factors are numbers we multiply together to get another number
• Common factors are factors that two or more numbers share
• The greatest common factor is the largest of these shared factors

How to Find the GCF

There are several methods to find the GCF. The simplest way is to list all factors of each number and find the largest one they have in common.

Let's find the GCF of 12 and 18:

Methods for Finding GCF

Factor Method

This is the method we just learned - listing all factors of each number and finding the largest common one. It works well for smaller numbers.

Prime Factorization Method

For larger numbers, we can use prime factors:

1. Find the prime factors of each number
2. Identify the common prime factors
3. Multiply these common factors together

Example: Find GCF of 24 and 36
Prime factors of 24: 2 × 2 × 2 × 3
Prime factors of 36: 2 × 2 × 3 × 3
Common prime factors: 2, 2, 3
GCF = 2 × 2 × 3 = 12

Division Method

Also called the Euclidean Algorithm:

1. Divide the larger number by the smaller number
2. If remainder is 0, the divisor is the GCF
3. If not, replace the larger number with the divisor, and the divisor with the remainder
4. Repeat until remainder is 0

Example: Find GCF of 48 and 18
48 ÷ 18 = 2 with remainder 12
18 ÷ 12 = 1 with remainder 6
12 ÷ 6 = 2 with remainder 0
GCF = 6

GCF Examples

Let's practice with some examples:

Example 1: Find the GCF of 15 and 25
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Common factors: 1, 5
GCF = 5

Example 2: Find the GCF of 36 and 48 using prime factors
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
Common prime factors: 2, 2, 3
GCF = 2 × 2 × 3 = 12

Example 3: Find the GCF of 14, 21, and 35
Factors of 14: 1, 2, 7, 14
Factors of 21: 1, 3, 7, 21
Factors of 35: 1, 5, 7, 35
Common factors: 1, 7
GCF = 7

Real-world Example: Maria has 12 apples and 18 oranges. She wants to make identical fruit baskets with no fruit left over. What is the greatest number of baskets she can make?
Solution: Find GCF of 12 and 18 = 6 baskets
Each basket will have 12÷6=2 apples and 18÷6=3 oranges

GCF Practice Quiz

Test your understanding with these GCF questions. Choose the correct answer for each question.

Frequently Asked Questions

Here are answers to common questions about Greatest Common Factor:

Math Trivia

Discover interesting facts about numbers and factors: