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Section 1.9
Prime Numbers and Prime
Factorizations
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Objectives
o Understand the terms prime number and composite
number.
o Understand the Sieve of Eratosthenes. (Optional)
o Determine whether a number is prime.
o Find the prime factorization of a composite number.
o Find all of the factors of a composite number.
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Prime Numbers and Composite Numbers
Prime Number
A prime number is a counting number greater than 1
that has exactly two different factors (or divisors) —
itself and 1.
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Prime Numbers and Composite Numbers
Composite Number
A composite number is a counting number with more
than two different factors (or divisors).
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Example 1: Prime and Composite Numbers
The following numbers are prime.
2 has exactly two different factors, 1 and 2.
2
7 has exactly two different factors, 1 and 7.
7
11 has exactly two different factors, 1 and 11.
11
29 has exactly two different factors, 1 and 29.
29
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Example 1: Prime and Composite
Numbers (cont.)
The following numbers are composite.
12
1  12 = 12, 2  6 = 12, and 3  4 = 12. 1, 2, 3, 4,
6, and 12 are all factors of 12. Thus 12 has more
than two different factors.
1  33 = 33 and 3  11 = 33. So 1, 3, 11, and 33
33
are all factors of 33, and 33 has more than two
different factors.
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Determining Whether a Number is Prime
To Determine Whether a Number is Prime
Divide the number by progressively larger prime
numbers (2, 3, 5, 7, 11, and so forth) until:
1. The remainder is 0. This means that the prime
number is a factor and the given number is
composite; or
2. You find a quotient smaller than the prime divisor.
This means that the given number is prime because
it has no smaller prime factors.
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Example 2: Determining Whether a
Number is Prime
Is 605 a prime number?
Solution
The ones digit is 5. Therefore, 605 is divisible by 5 and
is not prime. The number 605 is a composite number
and
605 = 5  121 = 5  11  11.
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Example 3: Determining Whether a
Number is Prime
Is 103 a prime number?
Solution
Tests for 2, 3, and 5 fail. (The number 103 is not even;
1 + 0 + 3 = 4 and 4 is not divisible by 3; and the last digit
is not 0 or 5.)
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Example 3: Determine Whether a
Number is Prime
Divide by 7:
14
7 103
7
33
28
5
The quotient is greater
than the divisor.
Divide by 11:
9
11 103
99
4
The quotient is less
than the divisor, so we
are done.
The remainder is not 0.
The remainder is not 0.
The number 103 is prime.
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Example 4: Determine Whether a
Number is Prime
Is 221 prime or composite?
Solution
Tests for 2, 3, and 5 fail.
Divide by 7:
3 1 The quotient is greater
7 221 than the divisor.
21
11
7
4 The remainder is not 0.
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Example 4: Determine Whether a
Number is Prime (cont.)
Divide by 11:
Divide by 13:
20
11 221
22
01
0
1
17
13 221
13
91
91
0
The quotient is greater
than the divisor.
The remainder is not 0.
The remainder is 0.
The number 221 is composite.
Note: 221 = 13×17; that is 13 and 17 are factors of 221.
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Example 5: Application of Factors of
Counting Numbers
One interesting application of factors of counting
numbers (very useful in beginning algebra) involves
finding two factors whose sum is some specified
number. For example, find two factors of 70 such that
their product is 70 and their sum is 19.
Solution
1  70 = 70,
2  35 = 70,
5  14 = 70, 7  10 = 70.
The numbers we are looking for are 5 and 14 because
5  14 = 70 and 5 + 14 = 19.
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The Prime Factorization of a Composite Number
The Fundamental Theorem of Arithmetic
Every composite number has exactly one prime
factorization.
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The Prime Factorization of a Composite Number
To Find the Prime Factorization of a
Composite Number
1. Factor the composite number into any two factors.
2. Factor each factor that is not prime.
3. Continue this process until all factors are prime.
The prime factorization is the product of all of the
prime factors.
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Example 6: Finding the Prime Factorization
of a Composite Number
Find the prime factorization of 60.
Solution
60 = 6

10
=23  25
Since the last digit is 0, we know 10 is a factor.
6 and 10 can both be factored so that
each factor is a prime number. This is the
prime factorization of 60.
We can also start with a different set of factors.
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Example 6: Finding the Prime Factorization
of a Composite Number
60 = 3

= 3 
20
4  5
= 3  2  25
3 is prime, but 20 is not.
4 is not prime.
All factors are prime.
Writing the factors in order, we see the prime
factorization of 60 is 2  2  3  5 or, using exponents,
22  3  5.
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Example 7: Finding the Prime Factorization
of a Composite Number
Find the prime factorization of 72.
Solution
72 = 8

72 = 2
9
= 2 4  3  3
or
= 2

36
 6  6
= 2223  3
= 2  2323
= 23  32
= 23  32
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Completion Example 8: Finding the Prime
Factorization of a Composite Number
Find the prime factorization of 196.
Solution
196 = 2

98
= 2
7
49
 __
__
2  14
= 2
7
2
7
2  __
7  __
 __
22  72
= ______
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using exponents
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Finding All of the Factors of a Composite
Number
Factors of a Composite Number
The only factors (or divisors) of a composite number
are:
1. 1 and the number itself;
2. each prime factor; and
3. products formed by all combinations of the prime
factors (including repeated factors).
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Example 9: Finding the Factors of a
Composite Number
Find all the factors of 30.
Solution
Because 30 = 2  3  5, the factors are
a. 1 and the number itself, 30.
b. Each prime factor: 2, 3, 5.
c. Products of all combinations of the prime factors:
2  3 = 6, 2  5 = 10,
3  5 = 15.
Therefore, the only factors are 1, 2, 3, 5, 6, 10, 15, and
30.
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Example 10: Finding the Factors of a
Composite Number
Find all the factors of 140.
Solution
The prime factorization of 140 is as follows
140 = 14  10
= 272 5
= 2  2  5  7.
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Example 10: Finding the Factors of a
Composite Number (cont.)
The factors of 140 are
a. 1 and the number itself: 1 and 140.
b. Each prime factor: 2, 5, 7.
c. Products of all combinations of the prime factors:
2  2 = 4,
2  5 = 10,
2  7 = 14,
5  7 = 35,
2  2  5 = 20, 2  2  7 = 28, 2  5  7 = 70.
The factors are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and
140.
There are no other factors (or divisors) of 140.
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Practice Problems
Determine whether each number is prime or
composite.
1. 39
2. 79
3. 151
4. 143
5. Find two factors of 80 such that their product is 80
and their sum is 21.
Find the prime factorization of each number.
6. 42
7. 56
8. 230
9. Using the prime factorization of 63, find all of the
factors of 63.
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Practice Problem Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
composite
prime
prime
composite
5 and 16
2·3·7
23 · 7
2 · 5 · 23
1, 3, 7, 9, 21, 63
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