Greatest common factor
The greatest common factor (GCF) of two or more numbers is the largest whole number that divides each of them evenly. To find it, list the prime factors of each number and multiply the factors they share. For example, the prime factors of 420 are 2, 2, 3, 5, 7 and the prime factors of 1050 are 2, 3, 5, 5, 7, so the GCF is 2 x 3 x 5 x 7 = 210. The GCF is taught in Chapter 3 of Saxon Math Course 2 and is essential in 7th grade math for simplifying fractions, factoring expressions, and solving ratio problems.
Key Concepts
Property The greatest common factor (GCF) of two numbers is the greatest whole number that divides both numbers evenly. It can be found by identifying all the common prime factors of the two numbers and multiplying them together.
Examples $$\text{Factors of 420 are } 2, 2, 3, 5, 7; \text{ Factors of 1050 are } 2, 3, 5, 5, 7. \text{ GCF is } 2 \cdot 3 \cdot 5 \cdot 7 = 210.$$ $$\text{Factors of 90 are } 2, 3, 3, 5; \text{ Factors of 324 are } 2, 2, 3, 3, 3, 3. \text{ GCF is } 2 \cdot 3 \cdot 3 = 18.$$ $$\text{Factors of 540 are } 2, 2, 3, 3, 3, 5; \text{ Factors of 600 are } 2, 2, 2, 3, 5, 5. \text{ GCF is } 2 \cdot 2 \cdot 3 \cdot 5 = 60.$$.
Explanation Meet the GCF, the biggest boss factor two numbers have in common! Using prime factorization, just find all the prime team members shared by both numbers and multiply them. This single 'boss factor' can then simplify a fraction in one clean shot, making your life way easier.
Common Questions
What is the greatest common factor?
The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. It is found by identifying the common prime factors and multiplying them together.
How do you find the GCF of two numbers?
Write the prime factorization of each number, identify the prime factors they share, and multiply those shared factors. For 90 (2 x 3 x 3 x 5) and 324 (2 x 2 x 3 x 3 x 3 x 3), the shared factors are 2, 3, and 3, so GCF = 18.
What is the GCF used for in math?
The GCF is used to reduce fractions to lowest terms, simplify ratios, and factor algebraic expressions. It also helps find the simplest form of a ratio between two quantities.
What is the difference between GCF and LCM?
The GCF is the greatest factor shared by two numbers, while the LCM (least common multiple) is the smallest number that both numbers divide into. GCF deals with factors; LCM deals with multiples.
Can you use a factor tree to find the GCF?
Yes. Build a factor tree for each number to find their prime factorizations. Then circle the prime factors that appear in both trees and multiply them for the GCF.
Is GCF taught in 7th grade?
Yes. Saxon Math Course 2 covers the greatest common factor in Chapter 3, where students use prime factorization and listing methods to find the GCF of two or more numbers.