Algebra Class: Inverses and GCD
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Questions and Answers

What is the inverse of 8 modulo 15?

  • 8
  • 7
  • 1
  • 2 (correct)

Which of the following can also be considered an inverse of 8 modulo 15?

  • 4
  • 11
  • -3 (correct)
  • 17 (correct)

Why does the integer 5 not have an inverse modulo 15?

  • 5 is greater than 1.
  • 5 is not coprime to 15. (correct)
  • 5 equals 15.
  • 5 is less than 15.

What is the result of multiplying both sides of the congruence 3x ≡ 4 (mod 7) by -2?

<p>x ≡ -8 (mod 7) (D)</p> Signup and view all the answers

What is the gcd of 4620 and 101 as determined in the content?

<p>1 (D)</p> Signup and view all the answers

What is the final inverse of 101 modulo 4620 as obtained through the method described?

<p>1601 (D)</p> Signup and view all the answers

Which approach is used to systematically find inverses as per the given content?

<p>Euclidean algorithm and Bézout coefficients (A)</p> Signup and view all the answers

What does Bézout's identity express?

<p>The gcd can be written as a linear combination of two integers. (D)</p> Signup and view all the answers

If $a = 6$ and $b = 14$, what is the value of $s$ in the equation $gcd(6,14) = 6(s) + 14(t)$?

<p>-2 (D)</p> Signup and view all the answers

Which of the following describes the Extended Euclidean Algorithm?

<p>A procedure to find the gcd and express it as a linear combination. (C)</p> Signup and view all the answers

If $a | b$ and $b | c$, which statement is true?

<p>$a | c$ (B)</p> Signup and view all the answers

In the theorem regarding valid congruences, under what condition does dividing both sides by an integer maintain validity?

<p>The integer must be relatively prime to the modulus. (D)</p> Signup and view all the answers

What does $a mod m$ represent?

<p>The remainder when $a$ is divided by $m$ (B)</p> Signup and view all the answers

What is the result of the gcd for the numbers 252 and 198?

<p>18 (B)</p> Signup and view all the answers

What are Bézout coefficients?

<p>The integers used in the linear combination for the gcd. (C)</p> Signup and view all the answers

In the context of modular arithmetic, what is true if $a ot eq b mod m$?

<p>They yield different remainders when divided by $m$ (C)</p> Signup and view all the answers

Which step is NOT part of computing Bézout coefficients?

<p>Finding all prime factors of the two integers. (A)</p> Signup and view all the answers

What does it mean for integers $a_1, a_2, ..., a_n$ to be pairwise relatively prime?

<p>Every pair of integers has a $gcd$ of 1 (B)</p> Signup and view all the answers

In the Euclidean algorithm, what do the equations represent?

<p>Stepwise deductions leading to the gcd. (C)</p> Signup and view all the answers

What is the result of the addition modulo $m$ operation denoted as $a +_m b$?

<p>The sum of $a$ and $b$ reduced to its least value modulo $m$ (B)</p> Signup and view all the answers

Which of the following methods is an efficient way to compute the greatest common divisor of two integers?

<p>The Euclidean algorithm (B)</p> Signup and view all the answers

How is the least common multiple found using prime factorization?

<p>By taking the highest exponent for all primes involved (C)</p> Signup and view all the answers

What happens to the congruence when adding two congruent integers modulo $m$?

<p>The sum modulo $m$ is the sum of the individual remainders (D)</p> Signup and view all the answers

What does the notation $a | b$ signify in number theory?

<p>$a$ is a divisor of $b$ (B)</p> Signup and view all the answers

In the Division Algorithm $b = qa + r$, which condition must hold for the remainder $r$?

<p>$r$ must be less than $a$ (C)</p> Signup and view all the answers

According to the properties of divisibility, if $a | b$ and $a | c$, what can be concluded?

<p>$a$ divides the sum $b + c$ (D)</p> Signup and view all the answers

What is the result of the expression $b ext{ mod } a$ if $b$ is exactly divisible by $a$?

<p>$0$ (C)</p> Signup and view all the answers

Which of the following statements is true regarding divisibility?

<p>If $a | b$, then some integer exists such that $b = aq$ (C)</p> Signup and view all the answers

What is the purpose of the Caesar cipher's encryption function?

<p>To shift letters in the alphabet securely. (A)</p> Signup and view all the answers

What kind of transformation does the decryption function $f^{-1}(p) = (p - 3) \mod 26$ perform?

<p>It shifts letters backward in the alphabet. (B)</p> Signup and view all the answers

If the integer key $k$ for a shift cipher is set to 5, what would be the encrypted form of the letter represented by the number 22?

<p>2 (A)</p> Signup and view all the answers

What must be true for a function $f$ to be considered a bijection in an affine cipher?

<p>The greatest common divisor gcd(a, 26) must equal 1. (A)</p> Signup and view all the answers

How would you decrypt the message represented by the numbers 4, 23, 15 using a shift cipher with $k = 4$?

<p>By subtracting 4 from each number. (D)</p> Signup and view all the answers

What does the term 'shift ciphers' refer to?

<p>Ciphers that change the original letters' positions. (B)</p> Signup and view all the answers

Which of the following messages is correctly encrypted using the Caesar cipher with $k = 3$?

<p>HELLO -&gt; KHOOR (D)</p> Signup and view all the answers

When encrypting with a Caesar cipher, what happens to the letters 'X', 'Y', and 'Z'?

<p>They wrap around to the start of the alphabet. (D)</p> Signup and view all the answers

What integers satisfy the equation $x \equiv 6 \ (mod \ 7)$?

<p>6, 13, 20 (D)</p> Signup and view all the answers

Which mathematician posed a problem involving modular arithmetic in the first century?

<p>Sun-Tsu (C)</p> Signup and view all the answers

What does the Chinese Remainder Theorem guarantee for a system of linear congruences?

<p>A unique solution exists modulo the product of the moduli (A)</p> Signup and view all the answers

When applying the Chinese Remainder Theorem, what is the value of $m$ given $m_1 = 3$, $m_2 = 5$, and $m_3 = 7$?

<p>105 (A)</p> Signup and view all the answers

What is $M_1$, given $m = 105$ and $m_1 = 3$?

<p>35 (D)</p> Signup and view all the answers

What is the correct first step in the procedure for solving a system of linear congruences using the Chinese Remainder Theorem?

<p>Calculate the product of all moduli (C)</p> Signup and view all the answers

Which of the following correctly finds $y_k$, the inverse of $M_k$ modulo $m_k$?

<p>Finding $y_k$ such that $M_k \cdot y_k \equiv 1 \ (mod \ m_k)$ (A)</p> Signup and view all the answers

In the context of the Chinese Remainder Theorem, what does the sum $x = a_1 M_1 y_1 + a_2 M_2 y_2 + ... + a_n M_n y_n$ represent?

<p>The solution to the system of linear congruences (D)</p> Signup and view all the answers

Flashcards

The Division Algorithm

The process of dividing one integer (the dividend) by another integer (the divisor) and obtaining a quotient and a remainder.

Divisibility

An integer 'a' divides another integer 'b' if there exists an integer 'q' such that b = aq. In other words, the remainder when dividing b by a is zero.

Property i: Sum of Divisibles

If 'a' divides 'b' and 'a' divides 'c', then 'a' also divides the sum of 'b' and 'c'.

Property ii: Product of Divisible

If 'a' divides 'b', then 'a' also divides the product of 'b' and any integer 'c'.

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GCD and LCM

The Greatest Common Divisor (GCD) of two integers is the largest integer that divides both of them. The Least Common Multiple (LCM) of two integers is the smallest integer that is a multiple of both of them.

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Greatest Common Divisor (GCD)

The largest integer that divides both 'a' and 'b'.

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Least Common Multiple (LCM)

The smallest integer that is a multiple of both 'a' and 'b'.

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Transitive Property of Divisibility

States that if 'a' divides 'b' and 'b' divides 'c', then 'a' must also divide 'c'.

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Congruence modulo m

A relationship between two integers 'a' and 'b' where they have the same remainder when divided by a positive integer 'm'.

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Zm (set of integers modulo m)

A set of non-negative integers less than 'm' which are used to represent remainders in modular arithmetic.

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Euclidean Algorithm

An efficient method for calculating the Greatest Common Divisor (GCD) of two integers.

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Pairwise Relatively Prime

A set of integers where the GCD of any pair of integers from the set is 1.

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Bézout's Identity

The greatest common divisor (GCD) of two integers, 'a' and 'b', can be expressed as a linear combination of 'a' and 'b' using integer coefficients 's' and 't'. This means we can write gcd(a,b) = sa + tb.

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Extended Euclidean Algorithm

The Extended Euclidean Algorithm is a method to find the GCD of two integers, 'a' and 'b', and simultaneously determine the Bézout coefficients 's' and 't' in the equation gcd(a,b) = sa + tb.

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Dividing Congruence by Relatively Prime Integer

A congruence where both sides are divided by an integer that is relatively prime to the modulus (meaning they have no common divisors other than 1) will result in a valid congruence.

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Dividing Congruences: Theorem

For any positive integer 'm' and integers 'a', 'b', and 'c', if 'a ≡ b (mod m)' and 'c' is relatively prime to 'm' (sharing no common divisors other than 1), then 'ac ≡ bc (mod m)' is a valid congruence.

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System of Linear Congruences

A system of equations involving congruences, where each equation relates a variable to a specific residue class modulo a different modulus.

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The Chinese Remainder Theorem

A theorem that provides a solution for a system of linear congruences with pairwise relatively prime moduli. It guarantees a unique solution within the range of the product of the moduli.

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What is 'm' in the CRT?

The product of all moduli in a system of linear congruences.

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What is 'Mk' in the CRT?

For each modulus in the system of linear congruences, it's the product of all other moduli.

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What is 'yk' in the CRT?

An integer that, when multiplied by 'Mk', leaves a remainder of 1 when divided by 'mk'.

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What is 'x' in the CRT?

It's the unique solution to the system of linear congruences, calculated using the CRT formula.

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What does 'pairwise relatively prime' mean?

Two integers are relatively prime if their greatest common divisor is 1.

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Why is the solution 'unique modulo m'?

The Chinese Remainder Theorem has a unique solution modulo 'm', meaning that any two solutions will be congruent modulo 'm'.

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Inverse modulo

An integer that, when multiplied by the original number, results in a remainder of 1 when divided by the modulus. For example, the inverse of 8 modulo 15 is 2, because 8 * 2 = 16, which leaves a remainder of 1 when divided by 15.

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Finding Inverses using Euclidean Algorithm

Using the Euclidean algorithm to find the greatest common divisor (GCD) of two numbers can help find the inverse of one number modulo the other. By working backward through the steps of the algorithm, you can express the GCD as a linear combination of the original numbers, where the coefficient of the number whose inverse you're looking for is the inverse.

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Solving Linear Congruences

A linear congruence equation is an equation of the form ax ≡ b (mod m), where a, b, and m are integers, and x is an unknown integer. To solve this equation, you need to find all possible values of x that satisfy the congruence. Multiplying both sides of the congruence by the inverse of 'a' modulo 'm' helps solve for x.

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Solvability of Linear Congruences

The congruence ax ≡ b (mod m) has solutions if and only if the greatest common divisor of 'a' and 'm' divides 'b'. If the GCD of 'a' and 'm' is 1, then the congruence has a unique solution modulo 'm'.

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Linear Combination

A linear combination of two integers a and b is an expression of the form as + bt, where s and t are integers. Bézout's identity is a special case of a linear combination.

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Non-Existence of Inverse

If the greatest common divisor of two numbers 'a' and 'm' is not 1, then the number 'a' does not have an inverse modulo 'm'. This means there is no number that you can multiply by 'a' to get a remainder of 1 when divided by 'm'.

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Caesar Cipher

A method of encrypting messages by shifting each letter a fixed number of positions down the alphabet, wrapping around to the beginning if necessary.

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Caesar Cipher Decryption

The function used to decrypt a Caesar cipher, which shifts each letter back the same number of positions as the encryption function shifted it forward.

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Shift Cipher

A family of ciphers where each letter is shifted by an integer number of positions down the alphabet. This integer, denoted by 'k', is called the key.

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Key (in Shift Cipher

The number of positions a letter is shifted in a shift cipher, determining the specific encryption function.

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Affine Cipher

A more general type of cipher using a linear function with two coefficients (a and b) to encrypt messages.

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Bijection (Affine Cipher)

A condition that ensures a one-to-one mapping between plaintext and ciphertext in an affine cipher. This means each letter is uniquely encoded and can be decoded back to its original form.

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Condition for Bijection (Affine Cipher)

The greatest common divisor between 'a' and 26 in the affine cipher equation (ap+b mod 26) must be 1. This condition guarantees a bijection, ensuring a proper encryption/decryption process.

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Alphabetic to Numeric Conversion

The process of converting letters in a message to numbers representing their positions in the alphabet (A=0, B=1, ..., Z = 25)

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Study Notes

Number Theory Topics

  • Number theory covers concepts like divisibility, modular arithmetic, primes, congruence, and more.
  • A presentation of a last lecture covered divisibility, modular arithmetic, congruence modulo, prime, GCD and LCM and the Euclidean Algorithm.
  • The division algorithm includes quotient and remainder calculation.

Divisibility

  • Divisibility refers to whether one integer divides another with a remainder of zero.
  • For integers a and b, if a is not equal to zero and there is an integer q such that b = aq, then a divides b.
  • Properties of divisibility include those of Theorem 1: If a divides b and a divides c, then a divides (b + c), if a divides b then a divides bc for all integers c, and if a divides b and b divides c then a divides c.

Congruence Relation

  • Two integers a and b are congruent modulo m if m divides (a - b), written as a ≡ b (mod m).
  • This means a and b have the same remainder when divided by m.
  • Algebraic manipulations are important.
  • If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m) and ac ≡ bd (mod m).

Arithmetic Modulo m

  • Arithmetic modulo m involves working with integers modulo m in computations
  • Given a modulus m (a positive integer) and integers a and b
  • Calculation of a +mb requires evaluating (a + b) modulo m.
  • Multiplication modulo m requires evaluating (a * b) modulo m.

Finding GCD Using Prime Factorizations

  • The greatest common divisor (GCD) of two integers a and b can be found using their prime factorizations.
  • The primes are in both prime factorization and each exponent is non-negative.
  • The GCD is found by taking the minimum of the corresponding exponents for each prime factor in both factorizations.

Greatest Common Divisor

  • Pairwise relatively prime integers have a GCD of 1.

Finding LCM Using Prime Factorization

  • The least common multiple (LCM) is found by taking the maximum of the corresponding exponents for each prime factor in both factorizations.

Euclidean Algorithm

  • The Euclidean algorithm is a method to compute the greatest common divisor (GCD) of two integers.
  • Repeatedly divides (a, b); if a < b, swap a and b and the remainder is now c.
  • Steps involving remainders: r = a mod b (remainder), gcd(a,b) = gcd(b,r).

GCDs as Linear Combinations

  • Bézout's Theorem states that if a and b are positive integers, there exist integers s and t such that gcd(a, b) = sa + tb.
  • This is called Bézout's identity.
  • The GCD can be expressed as a linear combination of a and b.
  • The Euclidean algorithm is used to find s and t computationally.

Dividing Congruences by an Integer

  • Dividing both sides of a congruence by an integer may not always produce a valid congruence.
  • But dividing by an integer that is relatively prime to the modulus (gcd(c, m) = 1) does result in a valid congruence.

Linear Congruence

  • A linear congruence is an equation of the form ax ≡ b (mod m).
  • It finds all x that satisfy the equation.

Solving Linear Congruences

  • Using inverses of a (mod m)- If the inverse exists, multiply both sides of the congruence by the inverse to solve for x.
  • The congruence can be solved if an inverse of a modulo m exists.

Inverse Modulo Theorem

  • If a and m are relatively prime, a unique inverse exists (mod m).

System of Linear Congruences

  • The Chinese Remainder Theorem provides a method to solve systems of linear congruences.
  • The congruences are of the form where m1, m2, and mn are relatively prime positive integers.
  • Finding the solution involves finding inverses modulo mk.

Fermat's Little Theorem

  • Fermat's Little Theorem: If p is a prime and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p).
  • The theorem can be used to simplify large power calculations modulo p.

Cryptography

  • Cryptography is the process of transforming information, preserving the integrity and confidentiality.
  • Number theory is the mathematical basis for classical ciphers, which were developed to secure and protect information.

Caesar Cipher

  • Caesar cipher is a type of cryptographic method that shifts each letter in the alphabet by a fixed number of positions.
  • It is a known and simple shift cipher. (p + k) mod 26.

Shift Cipher

  • Shift cipher uses a fixed letter shift of k, the key.
  • Encryption and decryption functions involved.
  • Shifts are determined by an integer (k), the key.

Affine Ciphers

  • Affine ciphers are generalizations of shift ciphers.
  • Used function (ap + b) mod 26 for encryption with a and b as integers; must use gcd(a, 26)=1 to prevent non-bijections and keep the cipher usable. It is a bijective function.

Private Key Cryptography

  • Private Key Cryptography involves using the same key for encryption and decryption.

Public Key Cryptography

  • Public Key Cryptography uses different keys for encryption and decryption; encryption key (public) for use by anyone and the decryption key is kept private.

RSA Cryptography

  • RSA is a public key cryptography system using two keys (n, e) for encryption, and (n, d) for decryption.
  • Integers n and e are part of the public key, typically generated from large prime factors.
  • The prime factors and encryption exponent must be kept private and relatively prime to prevent cracking.

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Number Theory Fall 2023-2024 Lecture Notes (PDF)

Description

Test your understanding of inverses in modular arithmetic, the Extended Euclidean Algorithm, and properties of greatest common divisors (gcd). This quiz covers concepts related to finding inverses, Bézout's identity, and conditions for valid congruences. Challenge yourself with a mix of theoretical and practical questions!

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