Greatest Common Factor
The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers without a remainder. In Grade 6 Saxon Math Course 1, students list all factors of each number, identify shared factors, and select the greatest. The GCF of 20 and 30 is 10 because it appears in both factor lists. GCF has direct applications: it determines the largest number of identical groups in sharing problems and is used to reduce fractions to lowest terms in one step.
Key Concepts
Property The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder.
Examples For 12 and 18, the factors of 12 are $1, 2, 3, 4, 6, 12$ and the factors of 18 are $1, 2, 3, 6, 9, 18$. The GCF is $6$. For 6, 9, and 15, the factors of 6 are $1, 2, 3, 6$; factors of 9 are $1, 3, 9$; factors of 15 are $1, 3, 5, 15$. The GCF is $3$. For 10 and 15, the factors of 10 are $1, 2, 5, 10$ and the factors of 15 are $1, 3, 5, 15$. The GCF is $5$.
Explanation Imagine two numbers showing off their factor collections. The GCF is the biggest, coolest factor they BOTH have. To find it, you list all the factors for each number, find all the factors they have in common, and then pick the largest one from that shared group. It's the king of all common factors!
Common Questions
How do you find the GCF of 20 and 30?
Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Common factors: 1, 2, 5, 10. GCF = 10.
How is GCF used to simplify fractions?
Divide both numerator and denominator by their GCF to reduce the fraction to lowest terms in a single step.
A teacher has 20 pencils and 30 erasers. What is the maximum number of identical kits?
GCF(20, 30) = 10. She can make 10 kits, each with 2 pencils and 3 erasers.
Can the GCF be larger than either number?
No. The GCF is always less than or equal to the smaller of the two numbers.
What is the difference between GCF and LCM?
GCF is the largest shared factor; LCM is the smallest shared multiple. GCF is used to reduce fractions; LCM is used to find common denominators.