Podcast
Questions and Answers
What is the primary focus of number theory?
What is the primary focus of number theory?
- The study of integers and their properties (correct)
- The study of prime numbers
- The study of the set of positive rational numbers
- The study of real numbers and their applications
What arithmetic method is introduced in number theory for divisibility?
What arithmetic method is introduced in number theory for divisibility?
- Algebraic arithmetic
- Vector arithmetic
- Modulo arithmetic (correct)
- Euclidean arithmetic
Which algorithm is used to compute the greatest common divisors?
Which algorithm is used to compute the greatest common divisors?
- Lattice algorithm
- Euclidean algorithm (correct)
- Sieve method
- Newton's method
Which of the following is NOT a topic covered in BSSE 21043?
Which of the following is NOT a topic covered in BSSE 21043?
In which applications does number theory play a critical role?
In which applications does number theory play a critical role?
Which of the following best describes 'divisibility' in number theory?
Which of the following best describes 'divisibility' in number theory?
Which of these topics is part of the fundamentals covered in this course?
Which of these topics is part of the fundamentals covered in this course?
What is an example of a reference used in the course?
What is an example of a reference used in the course?
What does the notation gcd(a, b) represent?
What does the notation gcd(a, b) represent?
When a is divided by d, what indicates that a is divisible by d?
When a is divided by d, what indicates that a is divisible by d?
Which of the following integers are relatively prime?
Which of the following integers are relatively prime?
If a = 12 and b = -18, what is gcd(a, b)?
If a = 12 and b = -18, what is gcd(a, b)?
Which statement about the operators for modular arithmetic is false?
Which statement about the operators for modular arithmetic is false?
What is the greatest common divisor of 24 and 36?
What is the greatest common divisor of 24 and 36?
Which of the following correctly describes gcd(12, -16)?
Which of the following correctly describes gcd(12, -16)?
How can the greatest common divisor of two integers be found?
How can the greatest common divisor of two integers be found?
What is the relationship between the product of two positive integers and their gcd and lcm?
What is the relationship between the product of two positive integers and their gcd and lcm?
What method is described as more efficient for finding the greatest common divisor?
What method is described as more efficient for finding the greatest common divisor?
In the first step of calculating gcd(91, 287), what is the remainder when 287 is divided by 91?
In the first step of calculating gcd(91, 287), what is the remainder when 287 is divided by 91?
After dividing 91 by 14, what is the result of the division and the remainder?
After dividing 91 by 14, what is the result of the division and the remainder?
What is the final result for gcd(91, 287) after performing the Euclidean algorithm?
What is the final result for gcd(91, 287) after performing the Euclidean algorithm?
Which of the following statements about divisors and remainders from the Euclidean algorithm is true?
Which of the following statements about divisors and remainders from the Euclidean algorithm is true?
What final equation represents the gcd of the original numbers using the backward substitution method?
What final equation represents the gcd of the original numbers using the backward substitution method?
Why is the Euclidean algorithm preferred over direct computation for gcd?
Why is the Euclidean algorithm preferred over direct computation for gcd?
What does gcd(a, b) represent in the context of the equations provided?
What does gcd(a, b) represent in the context of the equations provided?
In the linear combination gcd(a, b) = sa + tb, what do s and t represent?
In the linear combination gcd(a, b) = sa + tb, what do s and t represent?
Which equation represents the first step in calculating gcd for the numbers 414 and 662?
Which equation represents the first step in calculating gcd for the numbers 414 and 662?
What is the last non-zero remainder when calculating gcd(414, 662)?
What is the last non-zero remainder when calculating gcd(414, 662)?
From the equations derived, what is s when gcd(414, 662) is expressed as a linear combination?
From the equations derived, what is s when gcd(414, 662) is expressed as a linear combination?
Which of the following equations correctly follows the process of working backwards from the Euclidean algorithm?
Which of the following equations correctly follows the process of working backwards from the Euclidean algorithm?
What is the significance of the equation r_n = sa + tb in the Euclidean algorithm?
What is the significance of the equation r_n = sa + tb in the Euclidean algorithm?
What form does the linear combination take in order to express the gcd of two integers?
What form does the linear combination take in order to express the gcd of two integers?
Upon finishing the calculations, what will always be the value of gcd(a, b)?
Upon finishing the calculations, what will always be the value of gcd(a, b)?
Which sequence correctly summarizes the steps using the division algorithm to find the gcd of 414 and 662?
Which sequence correctly summarizes the steps using the division algorithm to find the gcd of 414 and 662?
What is Bézout's identity in the context of the greatest common divisor?
What is Bézout's identity in the context of the greatest common divisor?
If $gcd(a, b) = s imes a + t imes b$, what does it indicate about the integers 𝑎 and 𝑏?
If $gcd(a, b) = s imes a + t imes b$, what does it indicate about the integers 𝑎 and 𝑏?
What are the integers 𝑠 and 𝑡 called when expressing the gcd as a linear combination?
What are the integers 𝑠 and 𝑡 called when expressing the gcd as a linear combination?
Under what condition are two integers considered relatively prime?
Under what condition are two integers considered relatively prime?
What does the equation $gcd(252, 198) = 4 imes 252 + (-5) imes 198$ represent?
What does the equation $gcd(252, 198) = 4 imes 252 + (-5) imes 198$ represent?
Which division from the Euclidean algorithm is critical for expressing $gcd(252,198)$ as a linear combination?
Which division from the Euclidean algorithm is critical for expressing $gcd(252,198)$ as a linear combination?
What is the significance of the value 18 in the context of $gcd(252, 198)$?
What is the significance of the value 18 in the context of $gcd(252, 198)$?
Which of the following statements is true regarding integers when using Bézout's theorem?
Which of the following statements is true regarding integers when using Bézout's theorem?
Study Notes
Course Overview
- Course Code: BSSE 21043 - Mathematics for Software Engineering II
- Instructor: Prof. N.G.J. Dias, Senior Professor, Department of Computer Systems Engineering, University of Kelaniya
- Main Topics:
- Number Theory
- Graph Theory
- Algebraic Structures
- Combinatorics Applications
Evaluation Criteria
- End of Semester Examination: 60%
- Mid Semester Examination: 20%
- Assignments (2): 20%
Number Theory
- Definition: Study of integers and their properties.
- Key Concepts:
- Divisibility and modular arithmetic.
- Greatest common divisor (gcd) and least common multiple (lcm).
- Divisibility: Integer ( a ) is divisible by positive integer ( d ) if ( a \mod d = 0 ).
Modular Arithmetic
- Operators: Different programming languages use various symbols for mod (e.g.,
mod,%,rem). - Operators may yield different results for negative integers, requiring careful usage.
Greatest Common Divisor (gcd)
- Definition: gcd of integers ( a ) and ( b ) is the largest integer ( d ) such that ( d|a ) and ( d|b ).
- Notation: ( \text{gcd}(a, b) ).
- Existence: gcd exists because the set of common divisors is non-empty and finite.
- Relatively Prime: Two integers ( a ) and ( b ) are relatively prime if ( \text{gcd}(a, b) = 1 ).
Finding gcd
- Common Method: List and identify positive common divisors.
- Example:
- ( \text{gcd}(24, 36) = 12 )
- ( \text{gcd}(17, 22) = 1 ) (relatively prime).
Euclidean Algorithm
- Purpose: Efficient method for computing gcd without prime factorization.
- Process: Repeatedly apply division. For integers ( a ) and ( b ):
- Divide ( a ) by ( b ).
- Replace ( a ) with ( b ) and ( b ) with the remainder.
- Repeat until remainder is zero; last non-zero remainder is gcd.
Example using the Euclidean Algorithm
- Find ( \text{gcd}(91, 287) ):
- Step 1: ( 287 = 91 \times 3 + 14 )
- Step 2: ( 91 = 14 \times 6 + 7 )
- Step 3: ( 14 = 7 \times 2 + 0 )
- Thus, ( \text{gcd}(91, 287) = 7 ).
Bézout's Theorem
- States that gcd can be expressed as a linear combination of integers:
- ( \text{gcd}(a, b) = sa + tb ) for some integers ( s ) and ( t ).
- Bézout Coefficients: Integers ( s ) and ( t ) satisfying Bézout's identity.
Corollary
- Positive integers ( a ) and ( b ) are relatively prime if and only if there exist integers ( s ) and ( t ) such that ( sa + tb = 1 ).
Applications
- Number theory and graph theory have critical applications in software engineering, particularly in algorithms and data structures.
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Description
Test your understanding of Number Theory as part of the BSSE 21043 course. This quiz covers key concepts such as divisibility, modular arithmetic, and the greatest common divisor. Challenge yourself and reinforce your knowledge in this essential topic for software engineering.
