Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) of two numbers is the largest number that divides both evenly with no remainder. To find the GCF, list all factors of each number and identify the largest factor they share — the GCF of 24 and 40 is 8, because 8 is the largest number that divides both. This Grade 7 math skill from Saxon Math, Course 2 is essential for simplifying fractions to lowest terms, factoring algebraic expressions, and solving problems involving equal sharing or grouping.
Key Concepts
Property The greatest common factor (GCF) of two numbers is the largest common factor of both numbers.
Examples Find the GCF of 24 and 40. Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}. Factors of 40: {1, 2, 4, 5, 8, 10, 20, 40}. The GCF is 8. Find the GCF of 15 and 25. Factors of 15: {1, 3, 5, 15}. Factors of 25: {1, 5, 25}. The GCF is 5.
Explanation Imagine two numbers each have a team of factors. The GCF is the most valuable player that's on both teams! To find this VIP (Very Important Part), you list all the factors for each number, find all the ones they share, and then pick the biggest one from that shared list.
Common Questions
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor of two numbers is the largest number that divides both without a remainder. For 24 and 40, the GCF is 8 because 8 divides both (24 divided by 8 = 3, 40 divided by 8 = 5).
How do I find the GCF of two numbers?
List all the factors of each number, then find the largest factor that appears in both lists. For GCF(15, 25): factors of 15 are 1,3,5,15; factors of 25 are 1,5,25; the GCF is 5.
How is GCF used to simplify fractions?
Divide both the numerator and denominator by their GCF. For example, to simplify 24/40, find GCF(24,40) = 8, then divide: 24 divided by 8 = 3 and 40 divided by 8 = 5, giving 3/5.
What is the difference between GCF and LCM?
The GCF (Greatest Common Factor) is the largest number that divides two numbers, while the LCM (Least Common Multiple) is the smallest number that both numbers divide into. GCF is used for simplifying, LCM for adding fractions.
When do students learn about the GCF?
GCF is typically introduced in Grade 5-6 and reviewed in Grade 7. Saxon Math, Course 2 covers it in Chapter 3 as a tool for fraction simplification.
What are common mistakes when finding the GCF?
Students sometimes stop at a common factor that is not the greatest. Make sure to check all factors and identify the LARGEST one in common.
Which textbook covers GCF for Grade 7?
Saxon Math, Course 2 covers the Greatest Common Factor in Chapter 3, presenting it alongside fraction simplification to show its practical application.