Whole Numbers
Copyright © Cengage Learning. All rights reserved.
1
S E C T I O N 1.7
Prime Factors and Exponents
Copyright © Cengage Learning. All rights reserved.
Objectives
1.
Factor whole numbers.
2.
Identify even and odd whole numbers, prime
numbers, and composite numbers.
3.
Find prime factorizations using a factor tree.
4.
Find prime factorizations using a division
ladder.
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Objectives
5.
Use exponential notation.
6.
Evaluate exponential expressions.
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Prime Factors and Exponents
In this section, we will discuss how to express whole
numbers in factored form.
The procedures used to find the factored form of a whole
number involve multiplication and division.
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1
Factor whole numbers
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Factor whole numbers
The statement 3  2 = 6 has two parts: the numbers that are
being multiplied and the answer.
The numbers that are being multiplied are called factors,
and the answer is the product.
We say that 3 and 2 are factors of 6.
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Example 1
Find the factors of 12.
Strategy:
We will find all the pairs of whole numbers whose product
is 12.
Solution:
The pairs of whole numbers whose product is 12 are:
1  12 = 12, 2  6 = 12,
and
3  4 = 12
In order, from least to greatest, the factors of 12 are 1, 2, 3,
4, 6, and 12.
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Factor whole numbers
In Example 1, we found that 1, 2, 3, 4, 6, and 12 are the
factors of 12. Notice that each of the factors divides 12
exactly, leaving a remainder of 0.
In general, if a whole number is a factor of a given number,
it also divides the given number exactly.
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Factor whole numbers
When we say that 3 is a factor of 6, we are using the word
factor as a noun. The word factor is also used as a verb.
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2
Identify even and odd whole numbers, prime
numbers, and composite numbers
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Identify even and odd whole numbers, prime numbers, and composite numbers
A whole number is either even or odd.
The three dots at the end of each list shown above indicate
that there are infinitely many even and infinitely many odd
whole numbers.
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Identify even and odd whole numbers, prime numbers, and composite numbers
Numbers that have only two factors, 1 and the number
itself, are called prime numbers.
Note that the only even prime number is 2. Any other even
whole number is divisible by 2, and thus has 2 as a factor,
in addition to 1 and itself.
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Identify even and odd whole numbers, prime numbers, and composite numbers
Also note that not all odd whole numbers are prime
numbers. For example, since 15 has factors of 1, 3, 5, and
15, it is not a prime number.
The set of whole numbers contains many prime numbers. It
also contains many numbers that are not prime.
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Example 4
a. Is 37 a prime number?
b. Is 45 a prime number?
Strategy:
We will determine whether the given number has only 1
and itself as factors.
Solution:
a. Since 37 is a whole number greater than 1 and its only
factors are 1 and 37, it is prime. Since 37 is not divisible
by 2, we say it is an odd prime number.
b. The factors of 45 are 1, 3, 5, 9, 15, and 45. Since it has
factors other than 1 and 45, 45 is not prime. It is an odd
composite number.
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3
Find prime factorizations
using a factor tree
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Find prime factorizations using a factor tree
Every composite number can be formed by multiplying a
specific combination of prime numbers. The process of
finding that combination is called prime factorization.
One method for finding the prime factorization of a number
is called a factor tree.
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Find prime factorizations using a factor tree
The factor trees shown below are used to find the prime
factorization of 90 in two ways.
1. Factor 90 as 9  10.
1. Factor 90 as 6  15.
2. Neither 9 nor 10 are
prime, so we factor
each of them.
2. Neither 6 nor 15 are
prime, so we factor
each of them.
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Find prime factorizations using a factor tree
3. The process is complete
when only prime numbers
appear at the bottom of
all branches.
3. The process is complete
when only prime numbers
appear at the bottom of
all branches.
Either way, the prime factorization of 90 contains one factor
of 2, two factors of 3, and one factor of 5.
Writing the factors in order, from least to greatest, the
prime-factored form of 90 is 2  3  3  5.
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Find prime factorizations using a factor tree
It is true that no other combination of prime factors will
produce 90.
This example illustrates an important fact about composite
numbers.
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Example 5
Use a factor tree to find the prime factorization of 210.
Strategy:
We will factor each number that we encounter as a product
of two whole numbers (other than 1 and itself) until all the
factors involved are prime.
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Example 5 – Solution
The prime factorization of 210 is 7  5  2  3.
Writing the prime factors in order, from least to greatest, we
have 210 = 2  3  5  7.
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Example 5 – Solution
cont’d
Check:
Multiply the prime factors. The product should be 210.
2357=657
Write the multiplication in horizontal form.
Working left to right, multiply 2 and 3.
= 30  7
Working left to right, multiply 6 and 5.
= 210
Multiply 30 and 7. The result checks.
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4
Find prime factorizations using
a division ladder
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Find prime factorizations using a division ladder
We can also find the prime factorization of a whole number
using an inverted division process called a division ladder.
It is called that because of the vertical “steps” that it
produces.
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Example 6
Use a division ladder to find the prime factorization of 280.
Strategy:
We will perform repeated divisions by prime numbers until
the final quotient is itself a prime number.
Solution:
It is helpful to begin with the smallest prime, 2, as the first
trial divisor.
Then, if necessary, try the primes 3, 5, 7, 11, 13, … in that
order.
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Example 6 – Solution
cont’d
Step 1
The prime number 2 divides 280 exactly.
The result is 140, which is not prime.
Continue the division process.
Step 2
Since 140 is even, divide by 2 again.
The result is 70, which is not prime.
Continue the division process.
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Example 6 – Solution
cont’d
Step 3
Since 70 is even, divide by 2 a third time.
The result is 35, which is not prime.
Continue the division process.
Step 4
Since neither the prime number 2 nor the
next greatest prime number 3 divide 35
exactly, we try 5.The result is 7, which
is prime. We are done.
Prime
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Example 6 – Solution
cont’d
The prime factorization of 280 appears in the left column of
the division ladder: 2  2  2  5  7.
Check this result using multiplication.
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5
Use exponential notation
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Use exponential notation
In Example 6, we saw that the prime factorization of 280 is
2  2  2  5  7.
Because this factorization has three factors of 2, we call 2 a
repeated factor.
We can use exponential notation to write 2  2  2 in a
more compact form.
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Use exponential notation
The exponent is 3.
222 =
Repeated factors
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Read 23 as “2 to the third power” or “2 cubed.”
The base is 2.
The prime factorization of 280 can be written using
exponents: 2  2  2  5  7 = 23  5  7.
In the exponential expression 23, the number 2 is the
base and 3 is the exponent. The expression itself is called
a power of 2.
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Example 7
Write each product using exponents:
a. 5  5  5  5
b. 7  7  11
c. 2(2)(2)(2)(3)(3)(3)
Strategy:
We will determine the number of repeated factors in each
expression.
Solution:
a. The factor 5 is repeated 4 times. We can represent this
repeated multiplication with an exponential expression
having a base of 5 and an exponent of 4:
5  5  5  5 = 54
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Example 7 – Solution
cont’d
b. 7  7  11 = 72  11
7 is used as a factor 2 times.
c. 2(2)(2)(2)(3)(3)(3) = 24(33)
2 is used as a factor 4 times, and
3 is used as a factor 3 times.
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6
Evaluate exponential
expressions
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Evaluate exponential expressions
We can use the definition of exponent to evaluate (find the
value of) exponential expressions.
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Example 8
Evaluate each expression:
a. 72
b. 25
c. 104
d. 61
Strategy:
We will rewrite each exponential expression as a product of
repeated factors, and then perform the multiplication.
This requires that we identify the base and the exponent.
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Example 8 – Solution
We can write the steps of the solutions in horizontal form.
a.
72 =
77
= 49
b. 25 = 2  2  2  2  2
=4222
Read 72 as “7 to the second power” or “7
squared.” The base is 7 and the exponent
is 2. Write the base as a factor 2 times.
Multiply.
Read 25 as “2 to the 5th power.” The
base is 2 and the exponent is 5. Write
the base as a factor 5 times.
Multiply, working left to right.
= 8  2  2 = 16  2 = 32
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Example 8 – Solution
c. 104 = 10  10  10  10
= 100  10  10
cont’d
Read 104 as “10 to the 4th power.” The
base is 10 and the exponent is 4. Write the
base as a factor 4 times.
Multiply, working left to right.
= 1,000  10
= 10,000
d. 61 = 6
Read 61 as “6 to the first power.”
Write the base 6 once.
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