8.1 FACTORS AND GREATEST COMMON FACTORS

Write the prime factorization of numbers.

Find the GCF of monomials.

Objectives

prime factorizationgreatest common factor

Vocabulary

The whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors.

Factorizations of 12

Factorizations of 12

The circled factorization is the prime factorization because all the factors are prime numbers.

A prime number has exactly two factors, itself and 1.

Remember!

Example 1: Writing Prime Factorizations

Write the prime factorization of 98.

Method 1 Factor tree

Choose any two factorsof 98 to begin. Keep finding factors until each branch ends in a prime factor.

98

2 49

7 7

The prime factorization of 98 is 2 7 7

b. 40

Factors that are the same for two numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 32: 1, 2, 4, 8, 16, 32

Common factors: 1, 2, 4

The greatest of the common factors is 4.

Example 2A: Finding the GCF of Numbers

Find the GCF of each pair of numbers.100 and 60

factors of 100:

factors of 60:

List all the factors.

Circle the GCF.

Method 1 List the factors.

Example 3A: Finding the GCF of Monomials

Find the GCF of each pair of monomials.

15x3 and 9x2

15x3 = 3 5 x x x

9x2 = 3 3 x x

3 x x = 3x2

Write the prime factorization of each coefficient and write powers as products.

Align the common factors.

Find the product of the common factors.The GCF of 3x3 and 6x2 is 3x2.

Find the GCF of each pair of monomials.

8x2 and 7y3

If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power.

Helpful Hint