1. Objectives Write the prime factorization of numbers.
Find the GCF of monomials.

2. The whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors.
The prime factorization of a number is when you make a multiplication problem with only prime numbers that equals the number you are looking for.
Factorizations of 12
1 12 
2 6
3 4

3. Example 1: Writing Prime Factorizations
Write the prime factorization using a Factor Tree
a. 98
Choose any two factors
of 98. Keep finding factors until each branch ends in a prime factor.
b. 40 40  2 5 98 
40 = 23  5
The prime factorization of 40 is 2  2  2  5 or 23  5.
98 = 
The prime factorization of 98 is 2  7  7 or 2  72.

4. Factors that are shared by two or more whole numbers are called common factors. The biggest factor that both numbers have in common 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.
GCF = 4

5. Example 2A: Finding the GCF of Numbers
Find the GCF of each pair of numbers.
100 and 60
List the factors.
factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
List all the factors.
factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Circle the GCF.
The GCF of 100 and 60 is 20.

6. Check It Out! Example 2a
Find the GCF of each pair of numbers.
24 and 16
List the factors.
List all the factors.
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Circle the GCF.
factors of 16: 1, 2, 4, 8, 16
The GCF of 24 and 16 is 8.

7. Example 3A: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
Write the prime factorization of each coefficient and write powers as products.
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 15x3 and 9x2 is 3x2.

8. Example 3B: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime factorization of each coefficient and write powers as products.
8x2 = 2  2  2  x  x
7y3 = 7  y  y  y
Align the common factors.
There are no common factors other than 1.
The GCF 8x2 and 7y3 is 1.

9. Check It Out! Example 3c
Find the GCF of each pair of monomials.
8x and 4xv2
Write the prime factorization of each coefficient and write powers as products.
8x = 2  2  2  x
7v2 = 2  2  x  v  v
Align the common factors.
2  2  x = 4x
Find the product of the common factors.
The GCF of 8x and 4xv2 is 4x.

10. Check It Out! Example 3a
Find the GCF of each pair of monomials.
18g and 27g3
Write the prime factorization of each coefficient and write powers as products.
18g2 = 2  3  3  g 
27g3 =  3  3  g  g  g
Align the common factors.
3  3  g = 9g
Find the product of the common factors.
The GCF of 18g2 and 27g3 is 9g.