Slide 1 Copyright © 2015, 2011, 2008 Pearson Education, Inc. Factors and Prime Factorization...
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Transcript of Slide 1 Copyright © 2015, 2011, 2008 Pearson Education, Inc. Factors and Prime Factorization...
Slide 1Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Factors and Prime Factorization
Section 2.2
Slide 2Copyright © 2015, 2011, 2008 Pearson Education, Inc.
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.
Slide 3Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Examples
Find all the factors of each number.
a. 15
b. 7
c. 24
1, 3, 5, 15
1, 7
1, 2, 3, 4, 6, 8, 12, 24
Slide 4Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Prime and Composite Numbers
Prime Numbers
A prime number is a natural number that has exactly two different factors 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … .
Composite NumbersA composite number is a natural number, other than 1, that is not prime.
Slide 5Copyright © 2015, 2011, 2008 Pearson Education, Inc.
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.
Composite, it has more than two factors: 1, 7, 49.
Slide 6Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Prime Factorization
Every whole number greater than 1 has exactly one prime factorization.
Prime FactorizationThe prime factorization of a number is the factorization in which all the factors are prime numbers.
Slide 7Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Example
Find the prime factorization of 30.
Write 30 as the product of two numbers. Continue until all factors are prime.
30
6 • 5
3 • 2 • 5
The prime factorization of 30 is 2 · 3 · 5.
Slide 8Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Example
Find the prime factorization of 36.
Write 36 as the product of two numbers. Continue until all factors are prime.
36
9 • 4
3 • 3 2 • 2
The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22.
Slide 9Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Divisibility Tests
Slide 10Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Example
Write 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.
213 63
73 21
3 63
Slide 11Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Example
Find the prime factorization of 45.
Write 45 as the product of two numbers. Continue until all factors are prime.
45
9 • 5
3 • 3
The prime factorization of 45 is 3 · 3 · 5 or 32 · 5.
Slide 12Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Example
Find the prime factorization of 72.
Write 72 as the product of two numbers. Continue until all factors are prime.
72
9 • 8
3 • 3 4 • 2
2 • 2 3 22 2 2 3 3 or 2 3