Copyright © Cengage Learning. All rights reserved. 1 Whole Numbers

Copyright © Cengage Learning. All rights reserved.

1Whole Numbers

Copyright © Cengage Learning. All rights reserved.

S E C T I O N 1.7

Prime Factors and Exponents

33

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.

44

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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Factor whole numbers1

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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.

99

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.

1010

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.

1111

Identify even and odd whole numbers, prime numbers, and composite numbers

2

1212

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.

1313

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.

1414

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.

1515

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.

1616

Find prime factorizations using a factor tree

3

1717

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 2. Neither 6 nor 15 are

prime, so we factor prime, so we factor

each of them. each of them.

1919

Find prime factorizations using a factor tree

3. The process is complete 3. The process is complete

when only prime numbers when only prime numbers

appear at the bottom of appear at the bottom of

all branches. 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.

2222

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

Check:

Multiply the prime factors. The product should be 210.

2 3 5 7 = 6 5 7

= 30 7

= 210

cont’d

Write the multiplication in horizontal form.Working left to right, multiply 2 and 3.

Working left to right, multiply 6 and 5.

Multiply 30 and 7. The result checks.

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Find prime factorizations using a division ladder

4

2525

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

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.

cont’d

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Example 6 – Solution

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.

cont’d

Prime

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Example 6 – Solution

The prime factorization of 280 appears in the left column of the division ladder: 2 2 2 5 7.

Check this result using multiplication.

cont’d

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Use exponential notation5

3131

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

2 2 2 = 23

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.

Repeated factors The base is 2.

The exponent is 3.

Read 23 as “2 to the third power” or “2 cubed.”

3333

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

b. 7 7 11 = 72 11

c. 2(2)(2)(2)(3)(3)(3) = 24(33)

cont’d

7 is used as a factor 2 times.

2 is used as a factor 4 times, and 3 is used as a factor 3 times.

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Evaluate exponential expressions

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3636

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

= 8 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.

= 16 2

Multiply, working left to right.

= 32

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Example 8 – Solution

c. 104 = 10 10 10 10

= 100 10 10

= 1,000 10

= 10,000

d. 61 = 6

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.

Read 61 as “6 to the first power.” Write the base 6 once.

cont’d