Greatest common factor
Pre-algebra students in OpenStax Prealgebra 2E learn to find the greatest common factor (GCF) of two or more expressions by comparing their prime factorizations. The GCF of 24 and 36: factor to get 24 = 2^3 × 3 and 36 = 2^2 × 3^2; the GCF is 2^2 × 3 = 12. For 5x and 15: factor to get 5x = 5 × x and 15 = 3 × 5; the GCF is 5. Finding the GCF is the first step in factoring expressions, simplifying fractions, and solving problems involving divisibility.
Key Concepts
Property Splitting a product into factors is called factoring. The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
Examples To find the GCF of 24 and 36, we factor each number: $24 = 2 \cdot 2 \cdot 2 \cdot 3$ and $36 = 2 \cdot 2 \cdot 3 \cdot 3$. The common factors are $2 \cdot 2 \cdot 3$, so the GCF is 12.
To find the GCF of $5x$ and 15, we factor each term: $5x = 5 \cdot x$ and $15 = 3 \cdot 5$. The only common factor is 5.
Common Questions
What is the greatest common factor?
The GCF of two or more numbers is the largest factor that divides all of them evenly.
How do you find the GCF of 24 and 36?
Factor each: 24 = 2^3 × 3 and 36 = 2^2 × 3^2. Take the lowest power of each common prime: 2^2 × 3 = 12. The GCF is 12.
What is the GCF of 5x and 15?
5x = 5 × x and 15 = 3 × 5. The only common factor is 5, so GCF = 5.
How is the GCF used in simplifying fractions?
Divide numerator and denominator by their GCF to reduce the fraction to lowest terms.
How is the GCF used in factoring expressions?
Factor out the GCF from each term of an expression, writing it as a product: 24x + 36 = 12(2x + 3).