Greatest Common Factor (GCF)

Learn how to find the greatest common factor of two or more numbers using multiple methods.

Intermediate25 minLesson
Printable worksheet

Definition

The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides evenly into all of them.
For example, for the numbers 1212 and 1818:
  • Factors of 1212: 1,2,3,4,6,121, 2, 3, 4, 6, 12
  • Factors of 1818: 1,2,3,6,9,181, 2, 3, 6, 9, 18
  • Common factors: 1,2,3,61, 2, 3, 6
  • GCF = 6 (the largest common factor)
The GCF is also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

Try it now

What is the GCF of 66 and 99?

Worked Examples

Find the GCF of 2424 and 3636.
1

List all factors of 24

24=1×24=2×12=3×8=4×624 = 1 \times 24 = 2 \times 12 = 3 \times 8 = 4 \times 6 → Factors: 1,2,3,4,6,8,12,241, 2, 3, 4, 6, 8, 12, 24
2

List all factors of 36

36=1×36=2×18=3×12=4×9=6×636 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6 → Factors: 1,2,3,4,6,9,12,18,361, 2, 3, 4, 6, 9, 12, 18, 36
3

Circle the common factors

Both lists contain: 1,2,3,4,6,121, 2, 3, 4, 6, 12 → Common factors found
4

Find the greatest

The largest number in both lists is 1212 → GCF(24,36)=12\text{GCF}(24, 36) = 12

Common Mistakes

Confusing GCF with LCM

Why it's wrong: GCF is about dividing (finding what goes INTO both numbers), while LCM is about multiplying (finding what both numbers go INTO).

Correct: GCF of 12 and 18 is 6 (divides both). LCM of 12 and 18 is 36 (both divide into it).

Forgetting to take the LOWEST power when using prime factorization

Why it's wrong: Students sometimes multiply all prime factors instead of only the common ones with lowest powers.

Correct: For 24×32^4 \times 3 and 22×3×52^2 \times 3 \times 5, use 222^2 (not 242^4) and 313^1. GCF = 22×3=122^2 \times 3 = 12.

Stopping before finding the GREATEST common factor

Why it's wrong: Finding any common factor doesn't mean it's the greatest one.

Correct: For 24 and 36, common factors are 1, 2, 3, 4, 6, 12. The GCF is 12, not just any common factor like 2 or 6.

Interactive Visual

Venn Diagram - GCF & LCM

Number A:
Number B:
121822, 33Only in ACommonOnly in B

Prime factors of 12:

223

Prime factors of 18:

233

GCF

6

2 × 3 = 6

LCM

36

2 × 2 × 3 × 3 = 36

How it works:

  • • GCF = product of common prime factors (center of diagram)
  • • LCM = product of all prime factors (entire diagram)

Find the greatest common factor using prime factorization.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the GCF of 66 and 99?

Why It Matters

Finding the GCF is useful in many everyday situations:
  • Simplifying fractions: To reduce 1218\frac{12}{18} to lowest terms, divide by GCF(12, 18) = 6 to get 23\frac{2}{3}
  • Making equal groups: With 24 red tiles and 36 blue tiles, GCF(24, 36) = 12 is the largest number of identical groups with no tiles left over (each group gets 2 red and 3 blue tiles)
  • Cutting without waste: To cut boards of 48 cm and 60 cm into equal pieces with no waste, the longest possible piece is GCF(48, 60) = 12 cm
  • Tiling: A floor of 120 cm by 90 cm can be covered exactly with square tiles of side GCF(120, 90) = 30 cm, the largest such tiles
The GCF is the foundation for working with fractions and algebraic expressions!

Real World Applications

Simplifying Fractions

Reducing fractions to lowest terms requires finding the GCF of the numerator and denominator.

Example:

To simplify 4256\frac{42}{56}: GCF(42, 56) = 14, so 4256=42÷1456÷14=34\frac{42}{56} = \frac{42 \div 14}{56 \div 14} = \frac{3}{4}

1Try It Yourself

You need to simplify the fraction 3045\frac{30}{45}.

What is the simplified form?

Step 1: Choose the right answer

Divide 30 and 45 by GCF(30, 45). Which fraction do you get?

Cutting Materials Without Waste

When cutting boards or fabric, the GCF helps find the largest equal pieces possible.

Example:

You have a 72 cm board and a 90 cm board. To cut them into equal pieces with no waste, use pieces of GCF(72, 90) = 18 cm.

2Try It Yourself

You have ribbon rolls of 40 cm and 56 cm. You want to cut them into equal pieces with no leftover.

What is the longest piece you can cut?

Step 1: Write the mathematical expression

Find GCF(40, 56):

Organizing Items into Groups

The GCF helps when distributing different types of items into equal groups.

Example:

With 28 apples and 42 oranges, you can make GCF(28, 42) = 14 identical gift baskets, each with 2 apples and 3 oranges.

3Try It Yourself

A teacher has 36 pencils and 48 erasers to put into supply kits.

What is the maximum number of identical kits possible?

Step 1: Write the mathematical expression

Find GCF(36, 48):

Key Takeaways

  • 1The GCF is the largest number that divides evenly into two or more numbers
  • 2Method 1: List all factors of each number and find the greatest common one
  • 3Method 2: Use prime factorization and multiply the common primes with their lowest powers
  • 4The GCF is used to simplify fractions by dividing both numerator and denominator by it
  • 5When GCF = 1, the numbers are called relatively prime (no common factors besides 1)

Frequently Asked Questions

When GCF = 1, the numbers are called relatively prime or coprime. They share no common factors other than 1. Examples: GCF(8, 15) = 1, GCF(7, 12) = 1.
When GCF = 1, the numbers are called relatively prime or coprime. They share no common factors other than 1. Examples: GCF(8, 15) = 1, GCF(7, 12) = 1.
No, the GCF is always less than or equal to the smaller number. The GCF must divide both numbers, so it can't exceed either of them.
GCF is the largest number that divides INTO both numbers. LCM (Least Common Multiple) is the smallest number that both numbers divide INTO. For 12 and 18: GCF = 6, LCM = 36.

Glossary

Greatest Common Factor (GCF)
The largest positive integer that divides evenly into two or more numbers
Factor
A number that divides evenly into another number (no remainder)
Prime factorization
Writing a number as a product of prime numbers
Relatively prime
Two numbers whose GCF is 1 (also called coprime)
Common factor
A factor that is shared by two or more numbers
Today's challengeOne problem, three tries

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