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?
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