Untitled Quiz

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What relationship does the equation $ab = gcd(a, b) \cdot lcm(a, b)$ represent?

The ratio of $a$ to $b$

The relationship between GCD and LCM (correct)

The sum of $a$ and $b$

The difference between $a$ and $b$

If the $gcd(120, 500)$ is calculated, what can be inferred about the $lcm(120, 500)$?

It will be larger than 500

It is unrelated to GCD

It will be smaller than 120

It can be calculated using $gcd(120, 500)$ (correct)

What is the significance of using positive integers when defining the GCD and LCM relationship?

GCD and LCM are defined primarily for positive integers (correct)

Negative integers yield complex results

Negative integers do not exist in arithmetic

Positive integers are easier to compute

What is the least common multiple (LCM) of 120 and 500?

3000 Signup and view all the answers

Which statement is true about the LCM of two numbers?

It is the smallest number that both can divide Signup and view all the answers

How can the LCM of two numbers change if their GCD increases?

The LCM decreases Signup and view all the answers

What is the greatest common divisor (GCD) of 120 and 500?

20 Signup and view all the answers

In the calculation of LCM for 120 and 500, what is the smallest prime factor considered?

2 Signup and view all the answers

Which expression properly shows the relationship between GCD and LCM for any two numbers?

GCD(a, b) * LCM(a, b) = a * b Signup and view all the answers

What is the result of the LCM for the numbers 24 and 36 as calculated in the examples?

72 Signup and view all the answers

For the numbers 120 and 500, what is the calculation used to derive their GCD?

LCM(120, 500) / GCD(120, 500) Signup and view all the answers

In the prime factorization of 120, which prime numbers are included?

2, 3, 5 Signup and view all the answers

Regarding the definition of LCM, which concept is essential?

It is related to the divisibility of both numbers Signup and view all the answers

The prime factorization of 500 includes which of the following elements?

2, 5 Signup and view all the answers

What does LCM signify in relation to two numbers?

The smallest number that both numbers can divide into without a remainder Signup and view all the answers

When calculating the GCD and LCM, which method is used primarily?

Prime Factorization Method Signup and view all the answers

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

b is a multiple of a Signup and view all the answers

How can we express that a integer $a$ divides another integer $b$ using quantifiers?

∃c(a c = b) Signup and view all the answers

If $a | b$ and $a | c$, what can we conclude about $a$ in relation to $mb + nc$ where $m$ and $n$ are integers?

$a | mb + nc$ Signup and view all the answers

How do we determine the number of positive integers not exceeding $n$ that are divisible by a positive integer $d$?

By evaluating $n / d$ Signup and view all the answers

Given integers $a$, $b$, and $c$ where $a eq 0$, if $a | b$ and $b | c$, what can be inferred?

$a | c$ Signup and view all the answers

If $d$ is a positive integer, which of the following represents an integer that is not divisible by $d$?

$2d + 1$ Signup and view all the answers

What is the definition of a greatest common divisor (Gcd)?

The largest integer that divides given integers without leaving a remainder Signup and view all the answers

What condition must be met for an integer $a$ to divide the sum $b + c$ if $a | b$ and $a | c$?

a must not be zero Signup and view all the answers

Study Notes

Discrete Mathematics- Lecture 5

Topic: Number Theory
Chapter 4: Number Theory covers integers, integer representations, primes, greatest common divisors, and least common multiples.
Division Definition: If a and b are integers with a ≠ 0, 'a divides b' if there exists an integer c such that b = ac. a is a factor of b, b is a multiple of a. Notation: a | b.
Division (1/15): a divides b if there's an integer c such that b = ac. a is a factor of b, b is a multiple of a.
Division (2/15): Examples: 3 divides 12 because 12/3 = 4. 3 does not divide 7 because 7/3 is not an integer.
Division (3/15): A number line shows integers divisible by a positive integer d. d, 2d, 3d, ... represent integers divisible by d.
Example 3: Given positive integers n and d, the number of positive integers not exceeding n that are divisible by d is [n/d]
Theorem (Divisibility):
- If a|b and a|c, then a|(b+c)
- if a|b, then a|bc for all integers c
- if a|b and b|c, then a|c
- if a|b and a|c, then a|(mb + nc) for integers m and n
Division (7/15): The Division Algorithm:
- If a is an integer and d is a positive integer
- a = dq + r with 0 ≤ r < d
 - q = a div d (quotient)
 - r = a mod d (remainder)
 - The remainder cannot be negative
Example 1 (Division): Finding quotient and remainder when 101 is divided by 11. q = 9 and r = 2.
Example 2 (Division): Finding the quotient and remainder when -11 is divided by 3. q = -4 and r = 1.
Example 3 (Modular Arithmetic):
- 11 mod 2 = 1
- -11 mod 2 = 1
Note: If a|b, then -a|b
Example 4: Show that if a is an integer, then 1|a. q = a/1 = a , r = 0
Example 5: Demonstrates a|0 when a is not zero. q = 0/a = 0, r = 0
Example 6: Shows a|a, where a is an integer that is not zero. q = a/a = 1, r = 0
Example 7 (divisibility): If a|1, then a = ±1
Modular Arithmetic (Introduction 1/3): Remainders are important in certain situations, like timekeeping.
- (100 + 2) mod 24 = 6
  - time is 6:00
Modular Arithmetic (Definition): a is congruent to b modulo m, written a ≡ b (mod m), if m divides a − b.
Example 1 (Mod): Determine if integers are congruent to 5 modulo 6.
Example 2 (Mod): List five integers congruent to 2 modulo 4. (a = 2 + k*4)

Primes

Definition: A positive integer p > 1 is prime if its only positive factors are 1 and p.
Definition: A positive integer > 1 that is not prime is called composite.
Remark: The integer 1 is not prime. An integer n is composite if an integer a exists such that a | n and 1 < a < n.
Theorem 1 (Fundamental Theorem of Arithmetic): Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes.
Theorem 2: If n is a composite integer, then n has a prime divisor less than or equal to √n.
Examples:
- Is 100 Prime? No. (100 has prime divisors 2 and 5)
- Is 101 Prime? Yes. (101 has no prime divisors under √101 = 10).
- Prime factorization of 100: 2^2.5²
- Prime factorization of 1001: 7 * 11 * 13

The Sieve of Eratosthenes (1/6, 2/6, 3/6, ... 6/6)

Greatest Common Divisors (1/5)

Definition: The largest integer d such that d|a and d|b is called the greatest common divisor (gcd) of a and b.
- gcd(a,b) is notation
Example 1 (gcd): gcd(24, 36) = 12
Example 2 (gcd): gcd(120,500) = 20.
Definition 1: Integers a and b are relatively prime if their gcd is 1.
Example 1: 17 and 22 are relatively prime.
Definition 2: Integers a₁, a₂, ..., a_n are pairwise relatively prime if gcd(a_i, a_j) = 1 for all 1 ≤ i < j ≤ n.

Least Common Multiple (1/5, 2/5, ... 5/5)

Definition: The least common multiple (lcm) of positive integers a and b is the smallest positive integer that is divisible by both a and b.
Example 1 (lcm): lcm(24, 36) = 72
Example 2 (lcm): lcm(120, 500) = 3000
Theorem: ab = gcd(a, b) * lcm(a, b)
Example 3 (lcm and gcd): gcd(120, 500) and lcm(120, 500).

Applications (1/4, 2/4, 3/4, 4/4)

Hashing Functions: h(k) = k mod m
Pseudorandom numbers: Linear congruential method: X_n+1 = (a X_n + c) mod m
Cryptography: Encryption and decryption using modular arithmetic. Key is used.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Untitled Quiz

Choose a study mode

Podcast

Questions and Answers

What relationship does the equation $ab = gcd(a, b) \cdot lcm(a, b)$ represent?

If the $gcd(120, 500)$ is calculated, what can be inferred about the $lcm(120, 500)$?

What is the significance of using positive integers when defining the GCD and LCM relationship?

What is the least common multiple (LCM) of 120 and 500?

Which statement is true about the LCM of two numbers?

How can the LCM of two numbers change if their GCD increases?

What is the greatest common divisor (GCD) of 120 and 500?

In the calculation of LCM for 120 and 500, what is the smallest prime factor considered?

Which expression properly shows the relationship between GCD and LCM for any two numbers?

What is the result of the LCM for the numbers 24 and 36 as calculated in the examples?

For the numbers 120 and 500, what is the calculation used to derive their GCD?

In the prime factorization of 120, which prime numbers are included?

Regarding the definition of LCM, which concept is essential?

The prime factorization of 500 includes which of the following elements?

What does LCM signify in relation to two numbers?

When calculating the GCD and LCM, which method is used primarily?

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

How can we express that a integer $a$ divides another integer $b$ using quantifiers?

If $a | b$ and $a | c$, what can we conclude about $a$ in relation to $mb + nc$ where $m$ and $n$ are integers?

How do we determine the number of positive integers not exceeding $n$ that are divisible by a positive integer $d$?

Given integers $a$, $b$, and $c$ where $a eq 0$, if $a | b$ and $b | c$, what can be inferred?

If $d$ is a positive integer, which of the following represents an integer that is not divisible by $d$?

What is the definition of a greatest common divisor (Gcd)?

What condition must be met for an integer $a$ to divide the sum $b + c$ if $a | b$ and $a | c$?

Study Notes

Discrete Mathematics- Lecture 5

Primes

The Sieve of Eratosthenes (1/6, 2/6, 3/6, ... 6/6)

Greatest Common Divisors (1/5)

Least Common Multiple (1/5, 2/5, ... 5/5)

Applications (1/4, 2/4, 3/4, 4/4)

Studying That Suits You

Related Documents

More Like This

Untitled Quiz

Untitled Quiz

Untitled Quiz

Untitled Quiz