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Questions and Answers
What relationship does the equation $ab = gcd(a, b) \cdot lcm(a, b)$ represent?
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 $lcm(120, 500)$?
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?
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?
What is the least common multiple (LCM) of 120 and 500?
Which statement is true about the LCM of two numbers?
Which statement is true about the LCM of two numbers?
How can the LCM of two numbers change if their GCD increases?
How can the LCM of two numbers change if their GCD increases?
What is the greatest common divisor (GCD) of 120 and 500?
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?
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?
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?
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?
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?
In the prime factorization of 120, which prime numbers are included?
Regarding the definition of LCM, which concept is essential?
Regarding the definition of LCM, which concept is essential?
The prime factorization of 500 includes which of the following elements?
The prime factorization of 500 includes which of the following elements?
What does LCM signify in relation to two numbers?
What does LCM signify in relation to two numbers?
When calculating the GCD and LCM, which method is used primarily?
When calculating the GCD and LCM, which method is used primarily?
What does the notation $a | b$ represent in number theory?
What does the notation $a | b$ represent in number theory?
How can we express that a integer $a$ divides another integer $b$ using quantifiers?
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?
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 $d$?
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?
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 $d$?
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 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 $a | c$?
What condition must be met for an integer $a$ to divide the sum $b + c$ if $a | b$ and $a | c$?
Flashcards
Division Algorithm
Division Algorithm
Dividing an integer (dividend) by another positive integer (divisor) results in a quotient and remainder. The remainder is always less than the divisor.
Least Common Multiple (LCM)
Least Common Multiple (LCM)
The smallest positive integer that is a multiple of two or more integers.
Finding LCM
Finding LCM
To find the LCM, first find the prime factors of each number. The LCM is the product of the highest powers of each prime factor found in the numbers.
Relationship between LCM and GCD
Relationship between LCM and GCD
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GCD
GCD
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Prime factorization
Prime factorization
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Relatively prime
Relatively prime
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Example of LCM
Example of LCM
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What is the greatest common divisor (GCD)?
What is the greatest common divisor (GCD)?
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What is the least common multiple (LCM)?
What is the least common multiple (LCM)?
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How to find GCD
How to find GCD
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How to find LCM
How to find LCM
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GCD & LCM Relationship
GCD & LCM Relationship
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Hashing Functions
Hashing Functions
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Pseudorandom Numbers
Pseudorandom Numbers
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Cryptography
Cryptography
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a divides b
a divides b
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Divisibility
Divisibility
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Properties of Divisibility (1)
Properties of Divisibility (1)
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Properties of Divisibility (2)
Properties of Divisibility (2)
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Properties of Divisibility (3)
Properties of Divisibility (3)
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Divisibility and Linear Combinations
Divisibility and Linear Combinations
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Number of multiples of d not exceeding n
Number of multiples of d not exceeding n
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Integer Representations
Integer Representations
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Study Notes
Discrete Mathematics- Lecture 5
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Topic: Number Theory
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Chapter 4: Number Theory covers integers, integer representations, primes, greatest common divisors, and least common multiples.
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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.
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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.
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Division (2/15): Examples: 3 divides 12 because 12/3 = 4. 3 does not divide 7 because 7/3 is not an integer.
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Division (3/15): A number line shows integers divisible by a positive integer d. d, 2d, 3d, ... represent integers divisible by d.
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Example 3: Given positive integers n and d, the number of positive integers not exceeding n that are divisible by d is [n/d]
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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
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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
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Example 1 (Division): Finding quotient and remainder when 101 is divided by 11. q = 9 and r = 2.
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Example 2 (Division): Finding the quotient and remainder when -11 is divided by 3. q = -4 and r = 1.
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Example 3 (Modular Arithmetic):
- 11 mod 2 = 1
- -11 mod 2 = 1
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Note: If a|b, then -a|b
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Example 4: Show that if a is an integer, then 1|a. q = a/1 = a , r = 0
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Example 5: Demonstrates a|0 when a is not zero. q = 0/a = 0, r = 0
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Example 6: Shows a|a, where a is an integer that is not zero. q = a/a = 1, r = 0
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Example 7 (divisibility): If a|1, then a = ±1
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Modular Arithmetic (Introduction 1/3): Remainders are important in certain situations, like timekeeping.
- (100 + 2) mod 24 = 6
- time is 6:00
- (100 + 2) mod 24 = 6
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Modular Arithmetic (Definition): a is congruent to b modulo m, written a ≡ b (mod m), if m divides a − b.
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Example 1 (Mod): Determine if integers are congruent to 5 modulo 6.
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Example 2 (Mod): List five integers congruent to 2 modulo 4. (a = 2 + k*4)
Primes
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Definition: A positive integer p > 1 is prime if its only positive factors are 1 and p.
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Definition: A positive integer > 1 that is not prime is called composite.
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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.
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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.
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Theorem 2: If n is a composite integer, then n has a prime divisor less than or equal to √n.
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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: 22.52
- Prime factorization of 1001: 7 * 11 * 13
The Sieve of Eratosthenes (1/6, 2/6, 3/6, ... 6/6)
Greatest Common Divisors (1/5)
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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
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Example 1 (gcd): gcd(24, 36) = 12
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Example 2 (gcd): gcd(120,500) = 20.
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Definition 1: Integers a and b are relatively prime if their gcd is 1.
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Example 1: 17 and 22 are relatively prime.
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Definition 2: Integers a1, a2, ..., an are pairwise relatively prime if gcd(ai, aj) = 1 for all 1 ≤ i < j ≤ n.
Least Common Multiple (1/5, 2/5, ... 5/5)
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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.
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Example 1 (lcm): lcm(24, 36) = 72
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Example 2 (lcm): lcm(120, 500) = 3000
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Theorem: ab = gcd(a, b) * lcm(a, b)
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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: Xn+1 = (a Xn + c) mod m
- Cryptography: Encryption and decryption using modular arithmetic. Key is used.
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