Math 135 Midterm Exam-AID Session

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Nov 21, 2013 (3 years and 6 months ago)

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Math 135 Midterm Exam
-
AID Session

Agenda


2.3 Linear Diophantine Equations


2.5 Prime Numbers


3.1 Congruence


3.2 Tests for Divisibility


3.4 Modular Arithmetic


3.5 Linear
Congruences

Agenda


3.6 The Chinese Remainder Theorem


3.7 Euler Fermat Theorem


7.1
-
7.4 An Introduction to Cryptography


8.1
-
8.8 Complex Numbers

2.3 Linear Diophantine Equations


Definition

o
Linear Diophantine Equation

is an equation in
one or more unknowns with integer
coefficients, for which integer solutions are
sought


Linear Diophantine Equation Theorem



o

The linear Diophantine equation

has a
solution if and only if

2.3 Linear Diophantine Equations


Proposition



o

If
,

and



is one
particular solution, then the complete
integer solution is
for all
.

2.3 Linear Diophantine Equations


Process for Solving




o

Step 1: Find

o

Step 2: See if
; if true, continue. If false,
stop and state that the LDE has no solution

o

Step 3: Use
solution
found
in Extended Euclidean
Algorithm (or use Back
-
Substitution) to find a particular
solution to

o

Step 4: Multiply equation by

to get a particular
solution to

2.3 Linear Diophantine Equations


Process for Solving




o

Step 5: Use

for all

to
express the general form for all solutions to

o

Step 6: Apply Constraints (e.g. non
-
negativity) to
general solution

if necessary

2.3 Linear Diophantine Equations


Example




Example








o

Find all non
-
negative integer solutions to

o

Find all non
-
negative integer solutions to
14
𝑥

9
𝑢
=
1000

2.3 Linear Diophantine Equations


Example








o

A trucking company has to move 844 refrigerators. It
has two types of trucks it can use; one carries 28
refrigerators and the other 34 refrigerators. If it only
sends out full trucks and all the trucks return em
pty,
list the possible ways of moving all the refrigerators.

2.5 Prime Numbers


Definitions








o

A
d
ecimal system

is a set of

numbers
that
are written in
terms of powers of 10.

o

An integer

is c
alled a
prime

if its only positive
divisors are

and
; otherwise it’s called
composite
.

o

The
least common multiple

of two positive integers a
and b is the smallest positive integer that is divisible by
both a and b. It will be denoted by
.

2.5 Prime Numbers


Proposition 2.51



Euclid Theorem 2.52



Theorem 2.52



Unique Factorization Theorem 2.54







o

Every integer larger than 1 can be expressed as a
product of primes.

o

The number of primes is infinite.

o

If

is a prime and
, then

or
.

o

Every integer, greater than 1, can be expressed as a
product of primes and, apart from the order of the
factors, this expression is unique.

2.5 Prime Numbers


Theorem 2.55




Proposition 2.56







o

An

integer

is either prime or contains a prime
factor

o

If

is the prime factorization of a into
powers of distinct primes
, then the positive
divisors of

a
re those integers of the form

where

for

2.5 Prime Numbers


Theorem 2.57





Theorem 2.58












o

If


and
are
p
rime factorizations of the integers a and b, where
some of the exponents may be zero, then

where

for
.

o

If

and
are
prime factorizations of the integers a and b, where
some of the exponents may be zero, then

where

for
.

2.5 Prime Numbers


Example




Example



Example
















o

Factor
into prime factors and calculate
the greatest common divisor and least common
multiple or the two

numbers.

o

Prove that the sum of two consecutive odd pri
mes has
at least three prime divisors (not necessarily different)

o

Prove that

2.5 Prime Numbers


Example


















o

Let
, where

is a positive integer and

and

are odd primes. Prove that if

and
, then

or
.

3.1 Congruence


Definition





Proposition 3.11


















o

Let m be a fixed positive

integer. If
, we say
that “a is
congruent to

b modulo m” and write

whenever
. If
, we
write
.

o


o

If
, then

o

If

and
, then
.

3.1 Congruence


Example




















o

What is the remainder when

is divided by 7

3.2 Tests for Divisibility


Theorem 3.21

o
A number is divisible by 9 if and only if the sum
of its digits is divisible by 9.


Theorem 3.22

o
A number is divisible by 3 if and only if the sum
of its digits is divisible by 3.


Proposition 3.23

o
A number is divisible by 11 if and only if the
alternating sum of its digits is divisible by 11.




















3.2 Tests for Divisibility


Example




















o

Determine whether

is divisible by

3.4 Modular Arithmetic


Definitions




















o

The set of congruence classes of intege
rs, under the
congruence relation modulo m is called the set of
integers modulo m

and is denoted by

o

Modular arithmetic

is
given by
a
ddition and
multiplication
, and

are well defined in

o

The
congruence class modulo m

of the integer a is the
set integers




=

𝑥


|
𝑥



(
𝑚 𝑑

𝑚
)

.

3.4 Modular Arithmetic


Fermat’s Little Theorem




Corollary 3.43




















o

If p is a prime number that doesn’t divide the integer a,
then



1


1

(
𝑚 𝑑


)
.

o


For any intege
r a and prime p
,







(
𝑚 𝑑


)
.

3.4 Modular Arithmetic


Example



Example




















o

What is the remainder when

is divided by 7

o

Prove that

for all integers

3.5 Linear
Congruences


Definition



Linear Congruence Theorem 3.54




















o

A relation of the form

is called a
linear
congruence

in the variable x.

o

The one
-
variable linear congruence ax
c (mod m) has
a solution if and only if
.

o

If x
o

Z is one solution, then the complete solution is

where
.Hence there are

noncongruent solutions modulo
.

3.5 Linear
Congruences


Example



Example



Example



Example






















o

Find the inverse of

in

o

Determine the number of congruence classes in

that are solutions to the equation


o

Solve

o

Solve the congruence

1426
𝑥

597

(
𝑚 𝑑

805
)

3.6 The Chinese Remainder Theorem



Chinese Remainder Theorem 3.62

o

If
, then for any choice of
the integers

and
, the simultaneous
congruences



have a solution. Moreover, if
is one
integer solution, then the complete solution
is


3.6 The Chinese Remainder Theorem



Example

o

Solve the simultaneous congruences



3.6 The Chinese Remainder Theorem



Example

o

A basket contains a number of eggs and,
when the eggs are removed

at a time, there are

respectively, left over. When the eggs are
removed

at a time, there are none left
over. Assuming none of the eggs broke
during the preceding operations, determine
the minimum number of eggs there were in
the basket.

3.6 The Chinese Remainder Theorem



Example

o

A basket contains a number of eggs and,
when the eggs are removed

at a time, there are

respectively, left over. When the eggs are
removed

at a time, there are none left
over. Assuming none of the eggs broke
during the preceding operations, determine
the minimum number of eggs there were in
the basket.

3.7 Euler
-
Fermat Theorem



Euler
-
Fermat Theorem 3.71

o

If

is a positive integer and
,
then


7.1 Cryptography



Definition
s
:

o

C
ryptography
: study of sending message in a
secret or hidden form so that only those
people authorized to receive the m
essage
will be able to read i
t

o

Plaintext:
message being sent

o

Ciphertext:

encrypted message

7.2 Private Key Cryptography



Definition:

o

P
rivate
-
key system
: a method for data
encryption (and decryption) that requires
the parties who co
mmunicate to share a
common key

7.3 Public Key Cryptography



Definition
s
:

o

P
ublic
-
key cryptosystem
:
Each user has a
pair of
cryptographic keys



a
public

key and
a
private

key. The private key is kept secret,
whilst the public k
ey may be widely
distributed. Messages are
encrypted

with
the recipient's public key and can
only

be
decrypted with
the corresponding private
key

o

Public key
: refers to the encryption key

o

Private key
: decryption key

7.4 RSA Scheme

7.4 RSA Scheme

7.4 RSA Scheme

7.4 RSA Scheme

7.4 RSA Scheme



Ex
ample

o

Ron wants to send a message to Hermione
after encrypting it using RSA. Hermione’s
public key is

and her
private key is
. Using the
appropriate key, encrypt the message

that Ron wants to send to
Hermione.

7.4 RSA Scheme



Example

o

Suppose that
p
and
q
are prime numbers,

and

Prove that
and




Example

o

Suppose that

and
. Determine

and
.

8.1 Quadratic Equation



Quadratic Formula 8.11

o

If

then the quadratic equation
has the
solution

8.2 Complex Numbers



Definition:

o

C
omplex number
: an expression of the form
, where

The set of all
complex numbers is denoted by

8.2 Complex Numbers



Addition and Multiplication of Complex Numbers
8.21

o


o




Proposition 8.23

o


8.3 Complex Plane



Definition
s
:

o

Real axi
s:
The real axis is the line in the
complex plane

corres
ponding to zero
imaginary part

o

Imaginary axis
:
The axis in the
complex
plane

corresponding to zero
real part

o

Complex plane
: A

one
-
to
-
one
correspondence between th
e complex
numbers and the plane

8.3 Complex Plane



Definition
s
:

o

Modulus/absolute

value
:

T
he nonnegative
real number
If
then


o

C
omplex conjugate

of
is the
complex number

a complex
number multiplied by its conjugate
always

results in a real number

8.4 Properties of Complex Numbers



Proposition 8.42:

If z and w are complex numbers,
then

i.)


ii.)


iii.)


iv.)


v.)

is twice the real part of

vi.)

is 2i times the imaginary part of z.

8.4 Properties of Complex Numbers



Proposition 8.44: If z and w are complex numbers,
then

i.)


ii.)


iii.)


iv.)

(the triangle inequality)

8.4 Properties of Complex Numbers



Example

o

Write
in standard form.



Example

o

If
, prove that


8.4 Properties of Complex Numbers



Example

o

Shade the region of the complex plane for
which the following expression is real:


8.5 Polar Representation



Definition:

o

The
polar form

of the complex number
is
.

8.5 Polar Representation



Convert from Polar to Cartesian Coordinates 8.51

o


o


o

Conversely,
a point whose Cartesian
coordinates are (x,y) has the polar
coordinates

where
and
is an angle such
that

8.5 Polar Representation



Theorem 8.53

o

If
and
are two complex
numbers in polar form, then

8.5 Polar Representation



Example

o

Convert the numbers

and

to polar
form and multiply them together

8.6 De
Moivre’s

Theorem



C
omplex exponential function
:



De Moivre’s Theorem 8.61
:
For any real number




Corollary 8.62
:
If z=r(cos
then, for any
integer


8.6 De
Moivre’s

Theorem



Example

o

If

, express

in standard form.

8.7 Roots of Complex Numbers



Theorem 8.72
:

If
is the polar form
of a complex number, the
n all its complex nth roots
are equal to


The modulus
is the unique real nonnegative
root of

8.7 Roots of Complex Numbers



Example

o

Find all the solutions to

for

8.8 Fundamental Theorem of Algebra



Fundamental Theorem of Algebra 8.81

o

Every equation of the form
where
has at least one solut
ion in the
complex numbers.

Proofs to Memorize


Euclid’s Theorem 2.52


Proposition 2.53


Proposition 3.12


Proposition 3.14


Fermat’s Little Theorem 3.42


Proposition 7.41


Theorem 8.61






















Thanks!




Questions?