Download Chapter Two 2.2

2.2
Factors and Prime
Factorization
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
2
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:
1 6 and 2  3
So the factors of 6 are: 1, 2, 3 and 6.
Practice Problems 1
Find all the factors of each of the numbers.
115 and 3  5
1 7
c. 24 1 24, 2 12, 3  8 and 4  6
a. 15
b. 7
Answer is : 1, 3, 5, 15
Answer is : 1, 7
Answer is : 1, 2, 3, 4, 6, 8, 12, 24
P 122
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
3
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
pp 122-123
4
Examples
Determine whether each number is prime or composite.
Explain your answers.
a. 16
Composite, it has more than two factors: 1, 2, 4, 8, 16.
b. 31
Prime, its only factors are 1 and 31.
c. 49
Composite, it has more than two factors: 1, 7, 49.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
5
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.
21 13 18 29 39
13 29
21 18 39
p 123
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
6
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
p 123
7
Examples
Find the prime factorization of 63.
The first prime number 2 does not divide evenly, but 3 does.
21
3 63
Because 21 is not prime, we divide again.
7
3 21
3 63
The quotient 7 is prime, so we are finished. The prime
factorization of 63 is 3 · 3 · 7.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
8
Finding Prime Factorizations
Definition
Example
Prime Factorization: The
factorization in which all the
factors are prime numbers.
The prime factorization for 84.
42
2 84
21
2 42
7
3 21
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.
The prime factorizat ion for 84 is 22  3  7.
p 124
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
9
Factor Trees
Another way to find the prime factorization is to use
a factor tree.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
10
Finding Prime Factorizations
Definition
Prime Factorization:
The factorization in
which all the factors are
prime numbers.
Practice Problem 3
28
2
7
14
2
7
2 14
2 28
Answer is 2  2  7 or 2 2  7
p 125
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
11
Examples
Find the prime factorization of 30.
Write 30 as the product of two numbers. Continue until all
factors are prime.
30
6
3
• 2
•
5
• 5
The prime factorization of 30 is 2 · 3 · 5.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
12
Examples
Find the prime factorization of 36.
Write 36 as the product of two numbers. Continue until all
factors are prime.
36
9
3
• 3
•
4
2 • 2
The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
13
Finding Prime Factorizations
Definition
Practice Problem 4
Prime Factorization: The
factorization in which all
the factors are prime
numbers.
120
2
60
5
2
30
3 15
2
2 30
2 60
2 120
15
3
5
Answer is 2  2  2  3  5 or 23  3  5
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
p 124
14
Finding Prime Factorizations
Definition
Practice Problem 5
Prime Factorization:
The factorization in
which all the factors
are prime numbers.
7
756
126
6
2
3
2
63
3 21
3
21
3 63
3
3 189
2 378
2 756
7
Answer is 2  2  3  3  3  7 or 2 2  33  7
p 125
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
15
Finding Prime Factorizations
Definition
Prime Factorization:
The factorization in
which all the factors are
prime numbers.
Practice Problem 6
70
2
7
35
5
7
5 35
2 70
Answer is 2  5  7
p 125
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
16
Finding Prime Factorizations
Definition
Prime Factorization:
The factorization in
which all the factors are
prime numbers.
Practice Problem 7a
30
2
3
15
5
3
5 15
2 30
Answer is 2  3  5
p 126
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
17
Finding Prime Factorizations
Definition
Practice Problem 7b
Prime Factorization:
The factorization in
which all the factors are
prime numbers.
56
2
7
28
2
2 14
14
2
2 28
2 56
7
Answer is 2  2  2  7 or 23  7
Problem 7c: 72 and Problem 8: 117
p 126
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
18
DONE
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
19
Divisibility Tests
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Basic Mathematics, 4e
20