Download prime factorization

5.1
Number Theory
Copyright © 2009 Pearson Education, Inc.
Slide 5 - 1
Number Theory


The study of numbers and their properties.
The numbers we use to count are called natural
numbers, N, or counting numbers.
N  {1,2,3,4,5,...}
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Factors


The natural numbers that are multiplied together
to equal another natural number are called
factors of the product.
Example:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
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Slide 5 - 3
Divisors

If a and b are natural numbers and the quotient
of b divided by a has a remainder of 0, then we
say that a is a divisor of b or a divides b.
Copyright © 2009 Pearson Education, Inc.
Slide 5 - 4
Prime and Composite Numbers
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
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A prime number is a natural number greater
than 1 that has exactly two factors (or divisors),
itself and 1.
A composite number is a natural number that is
divisible by a number other than itself and 1.
The number 1 is neither prime nor composite, it
is called a unit.
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Slide 5 - 5
Rules of Divisibility
Divisible
Test
Example
by
2
The number is even.
846
3
The sum of the digits of
846
the number is divisible by since 8 + 4 + 6 = 18
3.
4
The number formed by
844
the last two digits of the
since 44  4
number is divisible by 4.
5
The number ends in 0 or
285
5.
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Slide 5 - 6
Create a list from 1 – 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
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3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
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The Fundamental Theorem of Arithmetic

Every composite number can be expressed as
a unique product of prime numbers.

This unique product is referred to as the prime
factorization of the number.
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Slide 5 - 8
Finding Prime Factorizations

Branching Method:

Select any two numbers whose product is
the number to be factored.

If the factors are not prime numbers,
continue factoring each number until all
numbers are prime.
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Slide 5 - 9
Example of branching method
3190
319
11
29
10
2
5
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29.
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Slide 5 - 10
Division Method
1. Divide the given number by the smallest prime
number by which it is divisible.
2. Place the quotient under the given number.
3. Divide the quotient by the smallest prime
number by which it is divisible and again
record the quotient.
4. Repeat this process until the quotient is a
prime number.
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Slide 5 - 11
Example of division method

Write the prime factorization of 663.
3 663
13 221
17

The final quotient 17, is a prime number, so
we stop. The prime factorization of 663 is
3 •13 •17
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Slide 5 - 12
Example 1: p. 218# 37
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Slide 5 - 13
Greatest Common Factor

The greatest common factor (GCF) of a set of
natural numbers is the largest natural number
that divides (without remainder) every number
in that set.
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Slide 5 - 14
Finding the GCF of Two or More
Numbers



Determine the prime factorization of each
number.
List each prime factor with smallest
exponent that appears in each of the prime
factorizations.
Determine the product of the factors found
in step 2.
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Slide 5 - 15
Example (GCF)
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Find the GCF of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7
Smallest exponent of each factor:
3 and 7
So, the GCF is 3 • 7 = 21.
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Slide 5 - 16
Find the GCF between 36 and 54
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Slide 5 - 17
Least Common Multiple

The least common multiple (LCM) of a set of
natural numbers is the smallest natural number
that is divisible (without remainder) by each
element of the set.
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Slide 5 - 18
Finding the LCM of Two or More
Numbers



Determine the prime factorization of each
number.
List each prime factor with the greatest
exponent that appears in any of the prime
factorizations.
Determine the product of the factors found in
step 2.
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Slide 5 - 19
Example (LCM)
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
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Find the LCM of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7
Greatest exponent of each factor:
32, 5 and 7
So, the LCM is 32 • 5 • 7 = 315.
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Slide 5 - 20
Find the LCM between 36 and 54
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Slide 5 - 21
Example of GCF and LCM


Find the GCF and LCM of 48 and 54.
Prime factorizations of each:
48 = 2 • 2 • 2 • 2 • 3 = 24 • 3
54 = 2 • 3 • 3 • 3 = 2 • 33
GCF = 2 • 3 = 6
LCM = 24 • 33 = 432
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Slide 5 - 22
Homework

P. 218# 15 – 54 (x3)
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