How to Factor the GCF
What Is the Greatest Common Factor?
The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers. It's also called the Greatest Common Divisor (GCD). Same thing, different name.
GCF shows up constantly in algebra. You'll need it for simplifying fractions, factoring polynomials, and solving Diophantine equations. It's not optional knowledge — it's foundational.
How to Find the GCF: Three Methods
Method 1: List All Factors
Write out every factor for each number, then find the biggest one they share.
Example: Find GCF of 36 and 48
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Common factors: 1, 2, 3, 4, 6, 12
- GCF = 12
This works fine for small numbers. For anything above 100, it gets tedious.
Method 2: Prime Factorization
Break each number into its prime factors. Then multiply the common primes.
Example: Find GCF of 72 and 108
- 72 = 2³ × 3²
- 108 = 2² × 3³
- Common primes: 2² (the smaller exponent) and 3² (the smaller exponent)
- GCF = 2² × 3² = 4 × 9 = 36
Method 3: The Ladder Method (Division by Primes)
Divide both numbers simultaneously by prime factors until no common primes remain.
Example: Find GCF of 84 and 126
2 | 84 126
3 | 42 63
7 | 14 21
2 3
Multiply the divisor numbers: 2 × 3 × 7 = 42
This method is fast and visual. Most students prefer it once they get comfortable.
Comparing the Three Methods
| Method | Best For | Speed | Works Well With |
|---|---|---|---|
| Listing Factors | Small numbers, beginners | Slow | Numbers under 50 |
| Prime Factorization | Algebraic expressions, variables | Medium | Any size numbers |
| Ladder Method | Speed, visual learners | Fast | Any size numbers |
Factoring Out the GCF from Expressions
This is where GCF becomes algebraically useful. When you have a polynomial expression, you can factor out the GCF to simplify it.
Example: Factor 12x² + 18x
- Find the GCF of the coefficients: GCF(12, 18) = 6
- Find the GCF of the variables: x² and x share x
- Combined GCF: 6x
- Divide each term by 6x: (12x² ÷ 6x) + (18x ÷ 6x) = 2x + 3
- Result: 6x(2x + 3)
Check by distributing: 6x × 2x = 12x². 6x × 3 = 18x. Correct.
More Complex Example
Factor: 8x³y² + 12x²y³ - 16xy⁴
- Coefficients: GCF(8, 12, 16) = 4
- x terms: x³, x², x → GCF = x
- y terms: y², y³, y⁴ → GCF = y²
- Overall GCF: 4xy²
- Factor out: 4xy²(2x² + 3xy - 4y²)
Common Mistakes to Avoid
- Forgetting the variables — students often find the GCF of coefficients only and ignore that x and y might be shared
- Using the largest exponent — wrong. Always use the smallest exponent when finding common primes
- Not checking your work — always distribute your factored answer to verify it matches the original
- Stopping too early — if terms still share a common factor, keep factoring
How to Get Started: Step-by-Step
Here's your workflow for any GCF problem:
- List or identify the coefficients. Find their GCF first.
- Look at the variables. Identify which ones appear in every term.
- Take the lowest power of each common variable.
- Multiply the coefficient GCF by the variable GCF.
- Divide each term by your GCF to get the remaining factor.
- Verify by distributing back.
Practice with numbers before jumping to variables. Master 48/84 before touching 12x²y and 18xy². Build the pattern recognition first.
When You'll Use This
GCF isn't just an isolated skill. It shows up in:
- Simplifying fractions — divide numerator and denominator by their GCF
- Factoring trinomials — GCF often factors out first before other methods
- Solving equations — factoring out GCF can reveal zero product property opportunities
- Adding/subtracting fractions — find common denominators using GCF
Every advanced algebra operation builds on this. Skip it and you'll struggle with everything that follows.