Discrete Structures

Chapter 4: Elementary Number Theory and Methods of Proof 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Be especially critical of any statement following the word “obviously”. – Anna Pell Wheeler, 1883-1966

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 1

Theorem 4.4.1 – The Quotient Remainder Theorem

Given any integer

n

and positive integer

d

,  unique integers

q

and

r

s.t.

n

=

dq

+

r

and 0 

r

<

d

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 2

Definitions

• Given an integer

n

and a positive integer

d

,

n div d

= the integer quotient obtained when

n

divided by

d

.

is

n mod d

= the nonnegative integer remainder obtained when

n

is divided by

d

.

Symbolically, if

n

and

d

are integers and

d

> 0, then

n div d

=

q

and

n mod d

=

r



n = dq

+

r

Where

q

and

r

are integers and 0 

r

<

d

.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 3

Example – pg. 189 # 8 & 9

• Evaluate the expressions.

8. a. 50

div

7 b. 50

mod

7 9. a. 28

div

5 b. 28

mod

5 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 4

Theorems

•

Theorem 4.4.2 – The Parity Property

Any two consecutive integers have opposite parity.

•

Theorem 4.4.3

The square of any odd integer has the form 8

m

+ 1 for some integer

m.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 5

Definition

• For any real number

x

, the

absolute value of

x

, is denoted |

x

|, is defined as follows:

x

  

x x

if

x

if

x

  0 0 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 6

Lemmas

•

Lemma 4.4.4

For all real numbers

r

, -|

r

| 

r

 |

r

| •

Lemma 4.4.5

For all real numbers

r

, |-

r

| = |

r

| •

Lemma 4.4.6 – The Triangle Inequality

For all real numbers

x

and

y

, |

x

+

y

|  |

x| +

|

y

| 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 7

Example – pg. 189 # 24

• Prove that for all integers

m

and

n

, if

m mod

5 = 2 and

n mod

5 = 1, then

mn mod

5 = 2.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 8

Example – pg. 190 #36

• Prove the following statement.

The product of any four consecutive integers is divisible by 8.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 9

Example – pg. 190 #42

• Prove the following statement.

Every prime number except 2 and 3 has the form 6

q

+ 1 or 6

q

+ 5 for some integer

q

.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 10

Example – pg. 190 #45

• Prove the following statement.

For all real numbers

r c



r



c

, then |

r

| 

c

.

and

c

with

c

 0, if 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 11

Example – pg. 190 # 49

• If

m

,

n

, and

d

are integers,

d

> 0, and

m mod d

that =

n mod d

, does it necessarily follow

m = n

? That

m

–

n

is divisible by

d

? Prove your answers.

4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem 12