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

• Definitiono Linear Diophantine Equation is an equation in

one or more unknowns with integer coefficients, for which integer solutions are sought

• Linear Diophantine Equation Theoremo The linear Diophantine equation has a

solution if and only if

2.3 Linear Diophantine Equations

• Propositiono If , and is one

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

2.3 Linear Diophantine Equations

• Process for Solvingo 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 Solvingo 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 14𝑥− 9𝑢 =1000

2.3 Linear Diophantine Equations

• Exampleo 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 empty, list the possible ways of moving all the refrigerators.

2.5 Prime Numbers

• Definitionso A decimal system is a set of numbers that are written in

terms of powers of 10. o An integer is called 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 are those integers of the form

where for

2.5 Prime Numbers

• Theorem 2.57

• Theorem 2.58

o If and are prime 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 primes has at least three prime divisors (not necessarily different)

o Prove that

2.5 Prime Numbers

• Exampleo 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

• Exampleo What is the remainder when is divided by 7

3.2 Tests for Divisibility

• Theorem 3.21o 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

• Exampleo Determine whether is divisible by

3.4 Modular Arithmetic

• Definitions

o The set of congruence classes of integers, under the congruence relation modulo m is called the set of integers modulo m and is denoted by

o Modular arithmetic is given by addition and multiplication, and are well defined in

o The congruence class modulo m of the integer a is the set integers ሾ𝑎ሿ= ሼ𝑥∈𝑍|𝑥≡ 𝑎 (𝑚𝑜𝑑 𝑚)ሽ.

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 𝑎𝑝−1 ≡ 1 (𝑚𝑜𝑑 𝑝).

o For any integer a and prime p, 𝑎𝑝 ≡ 𝑎 (𝑚𝑜𝑑 𝑝).

3.4 Modular Arithmetic

• Example

• Exampleo 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 xo Z is one solution, then the complete solution is where .Hence there are

noncongruent solutions modulo .

3.5 Linear Congruences

• Example

• Example

• Example

• Exampleo 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 1426𝑥≡ 597 (𝑚𝑜𝑑 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

Definitions: o Cryptography: study of sending message in a

secret or hidden form so that only those people authorized to receive the message will be able to read it

o Plaintext: message being sent

o Ciphertext: encrypted message

7.2 Private Key Cryptography

Definition:

o Private-key system: a method for data encryption (and decryption) that requires the parties who communicate to share a common key

7.3 Public Key Cryptography Definitions:

o Public-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 key 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

Example

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 Complex 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 Definitions:

o Real axis: The real axis is the line in the complex plane corresponding 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 the complex numbers and the plane

8.3 Complex Plane Definitions:

o Modulus/absolute value: The nonnegative real number Ifthen o Complex 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 andare 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

Complex 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, then 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 solution 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?