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2012 Pearson Prentice Hall. All rights reserved.

CHAPTER 3

Number Theory and theReal Number System

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2012 Pearson Prentice Hall. All rights reserved. 2

3.1

Number Theory: Prime & Composite

Numbers

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2012 Pearson Prentice Hall. All rights reserved.

Objectives

1. Determine divisibility.2. Write the prime factorization of a composite

number.

3. Find the greatest common divisor of twonumbers.

4. Solve problems using the greatest common

divisor.

5. Find the least common multiple of two numbers.

6. Solve problems using the least common

multiple.

3

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Number Theory and Divisibility

Number theory is primarily concerned with theproperties of numbers used for counting, namely 1, 2,

3, 4, 5, and so on.

The set of natural numbers is given by

Natural numbers that are multiplied together are

called thefactors of the resulting product.

,11,...6,7,8,9,101,2,3,4,5,N

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Divisibility

Ifa and b are natural numbers, a isdivisible by b if the

operation of dividing a by b leaves a remainder of 0.

This is the same as saying that b is adivisor ofa, or b

divides a.

This is symbolized by writing b|a.Example: We write 12|24 because 12 divides 24 or 24

divided by 12 leaves a remainder of 0. Thus, 24 is

divisible by 12.

Example: If we write 13|24, this means 13 divides 24 or

24 divided by 13 leaves a remainder of 0. But this is

not true, thus, 13|24.

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Aprime number is a natural number greater than 1

that has only itself and 1 as factors.

Acomposite number is a natural number greater than

1 that is divisible by a number other than itself and 1.

The Fundamental Theorem of ArithmeticEvery composite number can be expressed as a

product of prime numbers in one and only one

way.

One method used to find the prime factorization of a

composite number is called afactor tree.

Prime Factorization

E l 2 P i F i i i

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Example: Find the prime factorization of 700.Solution: Start with any two numbers whose product is

700, such as 7 and 100.

Example 2: Prime Factorization using aFactor Tree

Continue factoring the

composite number, branching

until the end of each branch

contains a prime number.

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Thus, the prime factorization of 700 is

700 = 7 2 2 5 5

= 7

2

2

5

2

Notice, we rewrite the prime factorization using a dot to

indicate multiplication, and arranging the factors from

least to greatest.

Example 2 (continued)

752 22

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Greatest Common Divisor

To find the greatest common divisor of two or more

numbers,

1. Write the prime factorization of each number.

2. Select each prime factor with the smallest exponentthat is common to each of the prime factorizations.

3. Form the product of the numbers from step 2. Thegreatest common divisor is the product of thesefactors.

Pairs of numbers that have 1 as their greatestcommon divisor are calledrelatively prime.

For example, the greatest common divisor of 5 and26 is 1. Thus, 5 and 26 are relatively prime.

E l 3 Fi di th G t t

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Example: Find the greatest common divisor of 216 and

234.

Solution:Step 1. Write the prime factorization of each

number.

Example 3: Finding the GreatestCommon Divisor

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216 = 23 33 234 = 2 32 13Step 2. Select each prime factor with the smallest

exponent that is common to each of the prime

factorizations.

Which exponent is appropriate for 2 and 3? We choose

the smallest exponent; for 2 we take 21, for 3 we take

32.

Example 3: (continued)

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Step 3. Form the product of the numbers from step 2.

The greatest common divisor is the product of thesefactors. Greatest common divisor = 2 32 = 2 9 =

18. Thus, the greatest common factor for 216 and 234

is 18.


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Least Common Multiple

The least common multiple of two or more natural

numbers is the smallest natural number that is divisibleby all of the numbers.

To find the least common multiple using prime

factorization of two or more numbers:1. Write the prime factorization of each number.

2. Select every prime factor that occurs, raised to the

greatest power to which it occurs, in these

factorizations.

3. Form the product of the numbers from step 2. The least

common multiple is the product of these factors.

E l 5 Fi di th L t C

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Example: Find the least common multiple of 144 and300.

Solution:Step 1. Write the prime factorization of each

number.

144 = 24 32

300 = 22 3 52

Step 2. Select every prime factor that occurs, raised to

the greatest power to which it occurs, in thesefactorizations. 144 = 24 32

300 = 22 3 52

Example 5: Finding the Least CommonMultiple

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Step 3. Form the product of the numbers from step 2.The least common multiple is the product of these

factors.

LCM = 24 3252 = 16 9 25 = 3600

Hence, the LCM of 144 and 300 is 3600. Thus, the

smallest natural number divisible by 144 and 300 is

3600.

Example 5: continued

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Example 6: Solving a Problem Using the LeastCommon Multiple

A movie theater runs its films continuously. One

movie runs for 80 minutes and a second runs for 120

minutes. Both movies begin at 4:00 P.M. When will

the movies begin again at the same time?

Solution:We are asked to find when the movies will

begin again at the same time. Therefore, we are

looking for the LCM of 80 and 120. Find the LCM

and then add this number of minutes to 4:00 P.M.

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Begin with the prime factorization of 80 and 120:80 = 24 5

120 = 23 3 5

Now select each prime factor, with the greatestexponent from each factorization.

LCM = 24 3 5 = 16 3 5 = 240

Therefore, it will take 240 minutes, or 4 hours, for the

movies to begin again at the same time. By adding 4

hours to 4:00 P.M., they will start together again at 8:00

P.M.


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