Download Factoring Whole Numbers

2.2
Factoring Whole Numbers
2.2
OBJECTIVES
1. Find the factors of a whole number
2. Find the prime factorization for any number
3. Find the greatest common factor (GCF) of
two numbers
4. Find the GCF for a group of numbers
To factor a number means to write the number as a product of its whole-number factors.
Example 1
Factoring a Composite Number
Factor the number 10.
10 2 5
The order in which you write the factors does not matter, so
10 5 2 would also be correct.
Of course, 10 10 1 is also a correct statement. However,
in this section we are interested in factors other than 1 and
the given number.
Factor the number 21.
21 3 7
CHECK YOURSELF 1
Factor 35.
In writing composite numbers as a product of factors, there will be a number of different possible factorizations.
Example 2
Factoring a Composite Number
Find three ways to factor 72.
© 2001 McGraw-Hill Companies
NOTE There have to be at
least two different
factorizations, because a
composite number has factors
other than 1 and itself.
72 8 9
(1)
6 12 (2)
3 24 (3)
CHECK YOURSELF 2
Find three ways to factor 42.
We now want to write composite numbers as a product of their prime factors. Look
again at the first factored line of Example 2. The process of factoring can be continued until
all the factors are prime numbers.
137
138
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
Example 3
Factoring a Composite Number
NOTE This is often called a
factor tree.
72
8
24
9
33
4 is still not prime, and so we
continue by factoring 4.
72 is now written as a product
of prime factors.
22233
NOTE Finding the prime
factorization of a number will
be important in our later work
in adding fractions.
When we write 72 as 2 2 2 3 3, no further factorization is possible. This is called
the prime factorization of 72.
Now, what if we start with the second factored line from the same example, 72 6 12?
Example 3 (Continued)
Factoring a Composite Number
72
6
23
12
34
23322
Continue to factor 6 and 12.
Continue again to factor 4. Other
choices for the factors of 12 are
possible. As we shall see, the end
result will be the same.
No matter which pair of factors you start with, you will find the same prime factorization. In this case, there are three factors of 2 and two factors of 3. Because multiplication
is commutative, the order in which we write the factors does not matter.
CHECK YOURSELF 3
We could also write
72 2 36
Continue the factorization.
Rules and Properties: The Fundamental Theorem
of Arithmetic
The method of the previous example will always work. However, an easier method for
factoring composite numbers exists. This method is particularly useful when numbers get
large, in which case factoring with a number tree becomes unwieldy.
Rules and Properties:
NOTE The prime factorization
is then the product of all the
prime divisors and the final
quotient.
Factoring by Division
To find the prime factorization of a number, divide the number by a series of
primes until the final quotient is a prime number.
© 2001 McGraw-Hill Companies
There is exactly one prime factorization for any composite number.
FACTORING WHOLE NUMBERS
SECTION 2.2
139
Example 4
Finding Prime Factors
To write 60 as a product of prime factors, divide 2 into 60 for a quotient of 30. Continue to
divide by 2 again for the quotient 15. Because 2 won’t divide evenly into 15, we try 3.
Because the quotient 5 is prime, we are done.
NOTE Do you see how the
divisibility tests are used here?
60 is divisible by 2, 30 is
divisible by 2, and 15 is divisible
by 3.
30
260
15
230
5
315
Prime
Our factors are the prime divisors and the final quotient. We have
60 2 2 3 5
CHECK YOURSELF 4
Complete the process to find the prime factorization of 90.
45
290
?
?45
Remember to continue until the final quotient is prime.
Writing composite numbers in their completely factored form can be simplified if we
use a format called continued division.
Example 5
Finding Prime Factors Using Continued Division
Use the continued-division method to divide 60 by a series of prime numbers.
NOTE In each short division,
we write the quotient below
rather than above the dividend.
This is just a convenience for
the next division.
Primes
2B60
2B30
3B15
5
Stop when the final quotient is prime.
To write the factorization of 60, we list each divisor used and the final prime quotient.
In our example, we have
60 2 2 3 5
CHECK YOURSELF 5
© 2001 McGraw-Hill Companies
Find the prime factorization of 234.
We know that a factor or a divisor of a whole number divides that number exactly.
The factors or divisors of 20 are
1, 2, 4, 5, 10, 20
NOTE Again the factors of 20,
other than 20 itself, are less
than 20.
Each of these numbers divides 20 exactly, that is, with no remainder.
Our work in this section involves common factors or divisors. A common factor or
divisor for two numbers is any factor that divides both the numbers exactly.
140
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
Example 6
Finding Common Factors
Look at the numbers 20 and 30. Is there a common factor for the two numbers?
First, we list the factors. Then we circle the ones that appear in both lists.
Factors
20:
1 , 2 , 4, 5 , 10 , 20
30:
1 , 2 , 3, 5 , 6, 10 , 15, 30
We see that 1, 2, 5, and 10 are common factors of 20 and 30. Each of these numbers divides
both 20 and 30 exactly.
Our later work with fractions will require that we find the greatest common factor (GCF)
of a group of numbers.
Definitions: Greatest Common Factor
The greatest common factor (GCF) of a group of numbers is the largest number
that will divide each of the given numbers exactly.
Example 6 (Continued)
Finding Common Factors
In the first part of Example 6, the common factors of the numbers 20 and 30 were listed as
1, 2, 5, 10
Common factors of 20 and 30
The greatest common factor of the two numbers is then 10, because 10 is the largest of the
four common factors.
CHECK YOURSELF 6
List the factors of 30 and 36, and then find the greatest common factor.
The method of Example 6 will also work in finding the greatest common factor of a
group of more than two numbers.
Finding the Greatest Common Factor (GCF) by Listing Factors
Find the GCF of 24, 30, and 36. We list the factors of each of the three numbers.
NOTE Looking at the three
24:
1 , 2 , 3 , 4, 6 , 8, 12, 24
lists, we see that 1, 2, 3, and 6
are common factors.
30:
1 , 2 , 3 , 5, 6 , 10, 15, 30
36:
1 , 2 , 3 , 4, 6 , 9, 12, 18, 36
6 is the greatest common factor of 24, 30, and 36.
© 2001 McGraw-Hill Companies
Example 7
FACTORING WHOLE NUMBERS
SECTION 2.2
141
CHECK YOURSELF 7
Find the greatest common factor (GCF) of 16, 24, and 32.
The process shown in Example 7 is very time-consuming when larger numbers are involved. A better approach to the problem of finding the GCF of a group of numbers uses
the prime factorization of each number. Let’s outline the process.
Step by Step: Finding the Greatest Common Factor
NOTE If there are no common
prime factors, the GCF is 1.
Step 1 Write the prime factorization for each of the numbers in the group.
Step 2 Locate the prime factors that are common to all the numbers.
Step 3 The greatest common factor (GCF) will be the product of all the
common prime factors.
Example 8
Finding the Greatest Common Factor (GCF)
Find the GCF of 20 and 30.
Step 1 Write the prime factorization of 20 and 30.
20 2 2 5
30 2 3 5
Step 2 Find the prime factors common to each number.
20 2 2 5
30 2 3 5
2 and 5 are the common prime factors.
Step 3 Form the product of the common prime factors.
2 5 10
10 is the greatest common factor.
CHECK YOURSELF 8
Find the GCF of 30 and 36.
© 2001 McGraw-Hill Companies
To find the greatest common factor of a group of more than two numbers, we use the
same process.
Example 9
Finding the Greatest Common Factor (GCF)
Find the GCF of 24, 30, and 36.
24 2 2 2 3
30 2 3 5
36 2 2 3 3
142
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
2 and 3 are the prime factors common to all three numbers.
2 3 6 is the GCF.
CHECK YOURSELF 9
Find the GCF of 15, 30, and 45.
Example 10
Finding the Greatest Common Factor (GCF)
15 and 28, have no common
factor other than 1, they are
called relatively prime.
Find the greatest common factor of 15 and 28.
15 3 5
28 2 2 7
There are no common prime factors
listed. But remember that 1 is a
factor of every whole number.
The greatest common factor of 15 and 28 is 1.
CHECK YOURSELF 10
Find the greatest common factor of 30 and 49.
CHECK YOURSELF ANSWERS
1. 5 7
2. 2 21, 3 14, 6 7
3. 2 2 2 3 3
4. 45
15
5
5. 2 3 3 13
290
345
315
90 2 3 3 5
6. 30: 1 , 2 , 3 , 5, 6 , 10, 15, 30
36: 1 , 2 , 3 , 4, 6 , 9, 12, 18, 36
6 is the greatest common factor.
7. 16: 1 , 2 , 4 , 8 , 16
24:
1 , 2 , 3, 4 , 6, 8 , 12, 24
32: 1 , 2 , 4 , 8 , 16, 32
The GCF is 8.
8. 30 2 3 5
36 2 2 3 3
The GCF is 2 3 6.
9. 15
10. GCF is 1; 30 and 49 are relatively prime
© 2001 McGraw-Hill Companies
NOTE If two numbers, such as
Name
2.2
Exercises
Section
Date
Find the prime factorization of each number.
1. 18
2. 22
ANSWERS
1.
2.
3. 30
4. 35
3.
4.
5.
5. 51
6. 42
6.
7.
7. 63
8. 94
8.
9.
10.
9. 70
10. 90
11.
12.
13.
11. 66
12. 100
14.
15.
16.
13. 130
14. 88
17.
18.
15. 315
16. 400
19.
© 2001 McGraw-Hill Companies
20.
17. 225
18. 132
19. 189
20. 330
143
ANSWERS
21.
In later mathematics courses, you often will want to find factors of a number with a given
sum or difference. The following exercises use this technique.
22.
23.
21. Find two factors of 24 with a sum of 10.
24.
25.
22. Find two factors of 15 with a difference of 2.
26.
27.
28.
23. Find two factors of 30 with a difference of 1.
29.
30.
24. Find two factors of 28 with a sum of 11.
31.
32.
33.
Find the greatest common factor (GCF) for each of the following groups of numbers.
34.
35.
25. 4 and 6
26. 6 and 9
27. 10 and 15
28. 12 and 14
29. 21 and 24
30. 22 and 33
31. 20 and 21
32. 28 and 42
33. 18 and 24
34. 35 and 36
35. 18 and 54
36. 12 and 48
144
© 2001 McGraw-Hill Companies
36.
ANSWERS
37. 12, 36, and 60
38. 15, 45, and 90
37.
38.
39.
39. 105, 140, and 175
40. 17, 19, and 31
40.
41.
41. 25, 75, and 150
42. 36, 72, and 144
42.
43.
43. A natural number is said to be perfect if it is equal to the sum of its counting number
44.
divisors, except itself.
45.
(a) Show that 28 is a perfect number.
46.
(b) Identify another perfect number less than 28.
44. Find the smallest natural number that is divisible by all of the following:
2, 3, 4, 6, 8, 9.
45. Tom and Dick both work the night shift at the steel mill. Tom has every sixth night
© 2001 McGraw-Hill Companies
off, and Dick has every eighth night off. If they both have August 1 off, when will
they both be off together again?
46. Mercury, Venus, and Earth revolve around the sun once every 3, 7, and 12 months,
respectively. If the three planets are now in the same straight line, what is the smallest
number of months that must pass before they line up again?
145
Answers
1. 2 3 3
11. 2 3 11
3. 2 3 5
13. 2B130
5. 3 17
7. 3 3 7
15. 3 3 5 7
9. 2 5 7
© 2001 McGraw-Hill Companies
5B65
13
130 2 5 13
17. 3 3 5 5
19. 3B189
21. 4, 6
23. 5, 6
25. 2
27. 5
3B63
3B21
7
189 3 3 3 7
29. 3
31. 1
33. 6
35. 18
37. 12
39. 35
41. 25
43.
45. August 25
146