Common Factoring- Finding the Greatest Common Factor

What Is the Greatest Common Factor?

The Greatest Common Factor (GCF), also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest number that divides two or more integers without leaving a remainder.

That's it. That's the whole concept.

For example, the GCF of 12 and 18 is 6 because 6 is the biggest number that goes into both 12 and 18 evenly.

GCF shows up constantly in math—simplifying fractions, factoring polynomials, and breaking down numbers into their basic building blocks. You need to know how to find it fast.

Three Methods to Find the GCF

There are three reliable ways to find the Greatest Common Factor. Pick whichever fits the situation.

Method 1: Listing All Factors

The most straightforward approach. Write out every factor of each number, then find the biggest one they share.

Example: Find GCF of 24 and 36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors: 1, 2, 3, 4, 6, 12

GCF = 12

This method works fine for small numbers. It falls apart when you're dealing with big integers or algebraic expressions.

Method 2: Prime Factorization

Break each number down into its prime factors, then multiply the common ones.

Example: Find GCF of 48 and 180

48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5

Common prime factors: 2² and 3

GCF = 2² × 3 = 4 × 3 = 12

This method is reliable and shows exactly why the GCF works. It's also the foundation for simplifying fractions and factoring polynomials.

Method 3: Euclidean Algorithm

This is the fast method for big numbers. You don't need to factor anything.

Steps:

• Divide the larger number by the smaller number
• Take the remainder
• Divide the previous divisor by the remainder
• Repeat until the remainder is 0
• The last non-zero remainder is your GCF

Example: Find GCF of 48 and 180

180 ÷ 48 = 3 remainder 36

48 ÷ 36 = 1 remainder 12

36 ÷ 12 = 3 remainder 0

GCF = 12

This algorithm handles huge numbers efficiently. Calculators use variations of this method.

GCF Method Comparison

Finding the GCF of Algebraic Expressions

The GCF of polynomials works the same way—you find the biggest factor that every term shares.

Example: Find the GCF of 6x³ and 9x²

6x³ = 2 × 3 × x × x × x

9x² = 3 × 3 × x × x

Common factors: 3, x²

GCF = 3x²

Example: Find the GCF of 12x²y and 18xy³

12x²y = 2 × 2 × 3 × x × x × y

18xy³ = 2 × 3 × 3 × x × y × y × y

Common factors: 2, 3, x, y

GCF = 6xy

The GCF of a polynomial is the product of the GCF of the coefficients and the GCF of the variable parts.

How to Factor Out the GCF (Getting Started)

Once you find the GCF, you can factor it out of an expression. This is called factoring out the GCF or using the distributive property in reverse.

Step 1: Find the GCF of all terms

Step 2: Divide each term by the GCF

Step 3: Write the GCF outside parentheses with the quotients inside

Example: Factor 12x + 18

GCF of 12x and 18 = 6

12x ÷ 6 = 2x

18 ÷ 6 = 3

Result: 6(2x + 3)

Example: Factor 8x³ + 12x² - 4x

GCF = 4x

8x³ ÷ 4x = 2x²

12x² ÷ 4x = 3x

-4x ÷ 4x = -1

Result: 4x(2x² + 3x - 1)

This process reverses the distributive property. It's the first step in factoring quadratics and solving polynomial equations.

Quick Practice Problems

Try these before checking the answers:

1. GCF of 16 and 24
2. GCF of 45, 60, and 75
3. GCF of 14x²y and 21xy²
4. Factor: 20 + 35
5. Factor: 6x² + 9x

Answers:

1. 8
2. 15
3. 7xy
4. 5(4 + 7)
5. 3x(2x + 3)

Where GCF Shows Up Next

Once you master finding and factoring out the GCF, you're ready for:

• Simplifying fractions — divide numerator and denominator by their GCF
• Factoring trinomials — GCF is always the first step before trying to factor x² + bx + c
• Adding/subtracting fractions — find the LCD by multiplying prime factors (related to GCF)
• Diophantine equations — finding integer solutions to ax + by = c

GCF isn't a standalone trick. It's a building block for almost everything in elementary algebra.