1. 8.1 Factors and Greatest Common Factors
Objectives
Write the prime factorization of numbers.
Find the GCF of monomials.
Vocabulary
prime factorization
greatest common factor

2. 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
1 12 
2 6
3 4
The circled factorization is the prime factorization because all the factors are prime numbers.
Factorizations of 12
1 12 
2 6
3 4
A prime number has exactly two factors, itself and 1.
Remember!

3. Example 1: Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
b. 40
Choose any two factors
of 98 to begin. Keep finding factors until each branch ends in a prime factor. 98 
The prime factorization of 98 is 2  7  7

4. Example 2A: Finding the GCF of Numbers
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
Method 1 List the factors.
factors of 100:
List all the factors.
factors of 60:
Circle the GCF.

5. Example 3A: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
Write the prime factorization of each coefficient and write powers as products.
15x3 and 9x2
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
Align the common factors.
3  x  x = 3x2
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