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1Copyright © 2009 Pearson Education, Inc.

Unit 1

Number Theory

MM-150 SURVEY OF MATHEMATICS – Jody Harris


Number Theory

Number theory is the study of natural numbers and their properties.

The numbers we use to count are called natural numbers, , or counting numbers.

{1,2,3,4,5,...}N

N


Factors and Divisors

The natural number a is a factor of b if there is another natural number k such that ak = b

Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Factors of a natural number are also called

divisors because if we divide a natural number by one if its factors the remainder is 0.


Prime and Composite Numbers A prime number is a natural number that has

exactly two factors (or divisors), itself and 1.



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; in other words it has more than 2 divisors.





The number 1 is neither prime nor composite, it is called a unit.





The number 1 is neither prime nor composite, it is called a unit.

Example: Which numbers are prime and which are composite? 2, 3, 9, 13, 22


Some Rules for Divisibility

8

2: Any even number, that is, any number that ends in 0, 2, 4, 6, or 8.

3: If the sum of its digits is divisible by 3, for example 2346 2+3+4+6 = 15, which is divisible by 3, so 2346 is.

5: If it ends in 0 or 5.

6: If it is divisible by both 2 AND 3, so 2346 is.

9: If the sum of its digits is divisible by 9, for example 2547 2+5+4+7 = 18, which is divisible by 9, so 2547 is.

10: If it ends in 0.


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.


Prime Factorization using a Factor Tree

2 32


Prime Factorization using a Factor Tree

2 32 23 3


Greatest Common Divisor The greatest common divisor (GCD) of a set of

natural numbers is the largest natural number that divides (without remainder) every number in that set.


Finding the GCD of Two or More Numbers Determine the prime factorization of each

number. List each common prime factor with

smallest exponent that appears in each of the prime factorizations.

The product of the factors found in the previous step are the GCD.


Example (GCD) Find the GCD of 63 and 105.

63 =



63 = 32 • 7 105 =



63 = 32 • 7 105 = 3 • 5 • 7



63 = 32 7 105 = 3 5 7

Smallest exponent of each common factor:3 and 7



63 = 32 • 7 105 = 3 • 5 • 7

Smallest exponent of each common factor:3 and 7

So, the GCD is 3 • 7 = 21.


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.


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.

The product of the factors found in step 2 is the LCM.


Example (LCM) Find the LCM of 63 and 105.

63 = 32 • 7105 = 3 • 5 • 7



63 = 32 • 7105 = 3 • 5 • 7

Greatest exponent of each factor:32, 5 and 7



63 = 32 • 7105 = 3 • 5 • 7

Greatest exponent of each factor:32, 5 and 7

So, the LCM is 32 • 5 • 7 = 315.


Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each:

48 =54 =



48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 =



48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33



48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33

GCD = 2 • 3 = 6



48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33

GCD = 2 • 3 = 6

LCM = 24 • 33 = 432


Whole Numbers

The set of whole numbers contains the set of natural numbers and the number 0.

Whole numbers = {0,1,2,3,4,…}


Integers The set of integers consists of 0, the natural

numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…} On a number line, the positive numbers extend

to the right from zero; the negative numbers extend to the left from zero.


The Rational Numbers• The set of rational numbers, denoted by Q,

is the set of all numbers of the form p/q, where p and q are integers and q 0.

• The following are examples of rational numbers:

13

,34

, 78

, 123

, 2, 0,157


Terminating or Repeating Decimal Numbers Every rational number when expressed as a

decimal number will be either a terminating or a repeating decimal number.

Examples of terminating decimal numbers are 0.7, 2.85, 0.000045

Examples of repeating decimal numbers 0.44444… which may be written

0.4, and 0.2323232323... which may be written 0.23.


Reducing Fractions

In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor.

Example: Reduce to its lowest terms. 7281


Reducing Fractions

In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor.

Example: Reduce to its lowest terms.

Solution:

7281

72 72 9 881 81 9 9


Multiplication of Fractions

Division of Fractions

, 0, 0a c a c ac b d

b d b d bd

, 0, 0, c 0a c a d ad b db d b c bc


Example: Multiplying Fractions Evaluate the following.

a)

b) 23

7

16

23

7

16

27316

1448

724

1

34

2

12

134

2

12

74

52

358

438


Example: Dividing Fractions Evaluate the following.a)

b) 23

67

23

67

23

76

2736

1418

79

58

45

58

45

58

54

5584

2532


Addition and Subtraction of Fractions

ac

bc

a b

c, c 0;

ac

bc

a b

c, c 0


Fundamental Law of Rational Numbers If a, b, and c are integers, with b 0, c 0,

then

ab

ab

cc

acbc

acbc

.


Example:

Evaluate:

Solution:

7

12

110

.

712

1

10

712

55

110

66

3560

660

2960


Irrational Numbers

An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number.

Examples of irrational numbers:

5.12639573...

6.1011011101111...0.525225222...


Radicals

are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

2, 17, 53


Product Rule for Radicals

Simplify:a)

b)

ab a b, a 0, b 0

40 410 4 10 2 10 2 10

125 255 25 5 5 5 5 5

40

125


Addition and Subtraction of Irrational Numbers

To add or subtract two or more square roots with the same radicand, add or subtract their coefficients.

The answer is the sum or difference of the coefficients multiplied by the common radical.


Example: Adding or Subtracting Irrational Numbers

Simplify: Simplify: 4 7 3 7

4 7 3 7

(4 3) 7

7 7

8 5 125

8 5 125

8 5 25 5

8 5 5 5

(8 5) 5

3 5


Multiplication of Irrational Numbers

Simplify:

6 54

6 54 654 324 18


Quotient Rule for Radicals

ab

ab

, a 0, b 0


Example: Division

Divide:

Solution:

Divide:

Solution:

16

4

144

2

16

4

164

4 2

144

2

1442

72

362 36 2

6 2


Rationalizing the Denominator A denominator is rationalized when it contains

no radical expressions. To rationalize the denominator, multiply BOTH

the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result.


Example: Rationalize

Rationalize the denominator of

Solution:

8

12.

8

12

812

23

2

3

2

3

3

3

6

3

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