1. Factors and Prime Factorization
2.2
Factors and Prime Factorization

2. Finding the Factors of Numbers
To perform many operations, it is necessary to be able to factor a number.
Since 7 · 9 = 63, both 7 and 9 are factors of 63, and 7 · 9 is called a factorization of 63.
Objective A 2

3. Finding Factors of Numbers
Definition
Example
Factors: When numbers are multiplied to form a product, each number is called a factor.
The different factorizations of 6 are:
So the factors of 6 are: 1, 2, 3 and 6.
Practice Problems 1
Find all the factors of each of the numbers.
Objective A
P 122

4. Prime and Composite Numbers
Prime Numbers
A prime number is a natural number that has exactly two different factors 1 and itself.
Composite Numbers
A composite number is any natural number, other than 1, that is not prime.
Objective A pp 4

5. Examples
Determine whether each number is prime or composite. Explain your answers.
a. 16
b. 31
c. 49
Composite, it has more than two factors: 1, 2, 4, 8, 16.
Prime, its only factors are 1 and 31.
Objective A
Composite, it has more than two factors: 1, 7, 49. 5

6. Identifying Prime and Composite Numbers
Definition
Example
Prime Number: A natural number that has exactly two different factors, 1 and itself.
The first several prime numbers are:
2, 3, 5, 7, 11, 13, 17
Composite Number: If a natural number other than 1 is not a prime number, it is called a composite number.
The number 10 has more than two factors:
1, 2, 5, and 10
Practice Problems 2
Determine whether each number is
Prime or
Composite.
Objective B
p 123

7. Prime Factorization Prime Factorization
The prime factorization of a number is the factorization in which all the factors are prime numbers.
Every whole number greater than 1 has exactly one prime factorization.
Objective A
p 123 7

8. Examples Find the prime factorization of 63.
The first prime number 2 does not divide evenly, but 3 does.
Because 21 is not prime, we divide again.
The quotient 7 is prime, so we are finished. The prime factorization of 63 is 3 · 3 · 7.
Objective A 8

9. Finding Prime Factorizations
Definition
Example
Prime Factorization: The factorization in which all the factors are prime numbers.
The prime factorization for 84.
because 42 is not a prime number we must divide it by a prime number.
because 21 is not a prime number we must divide it by a prime number.
because 7 is a prime number we can now write the prime factorization of 84.
Objective C
p 124

10. Factor Trees
Another way to find the prime factorization is to use a factor tree.
Objective A 10

11. Finding Prime Factorizations
Definition
Practice Problem 3
Prime Factorization: The factorization in which all the factors are prime numbers. 28 2 14 2 7
Objective C
p 125

12. Examples Find the prime factorization of 30.
Write 30 as the product of two numbers. Continue until all factors are prime. 30 •
3 • • 5
The prime factorization of 30 is 2 · 3 · 5.
Objective A 12

13. Examples Find the prime factorization of 36.
Write 36 as the product of two numbers. Continue until all factors are prime. 36 •
3 • • 2
The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22.
Objective A 13

14. Finding Prime Factorizations
Definition
Practice Problem 4
Prime Factorization: The factorization in which all the factors are prime numbers.
120 2 60 2 30 2 15
Objective C 3 5
p 124

15. Finding Prime Factorizations
Definition
Practice Problem 5
Prime Factorization: The factorization in which all the factors are prime numbers.
756
126 6 2 3 2 63 3 21 3 7
Objective C
p 125

16. Finding Prime Factorizations
Definition
Practice Problem 6
Prime Factorization: The factorization in which all the factors are prime numbers. 70 2 35 5 7
Objective C
p 125

17. Finding Prime Factorizations
Definition
Practice Problem 7a
Prime Factorization: The factorization in which all the factors are prime numbers. 30 2 15 5 3
Objective C
p 126

18. Finding Prime Factorizations
Definition
Practice Problem 7b
Prime Factorization: The factorization in which all the factors are prime numbers. 56 2 28 2 14 2 7
Objective C
Problem 7c: 72 and Problem 8: 117
p 126

19. DONE

20. Divisibility Tests
Objective A 20