Discrete Mathematics - Vol 3 | 9. Lecture – 57: Properties of GCD and Bezout’s Theorem by Abraham | Learn Smarter
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9. Lecture – 57: Properties of GCD and Bezout’s Theorem

The chapter discusses properties of the greatest common divisor (GCD) and Bezout’s theorem, emphasizing the expressibility of GCD as a linear combination of two integers. The extended Euclidean algorithm is introduced for determining GCD and finding Bezout's coefficients, which are crucial for calculating modular multiplicative inverses. Additionally, the conditions for the existence of modular inverses are outlined, focusing on coprimality between integers and their modulus.

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Sections

  • 9

    Lecture – 57: Properties Of Gcd And Bezout’s Theorem

    This lecture discusses the properties of GCD and Bezout’s theorem, demonstrating how the GCD of two integers can be represented as a linear combination of those integers.

    Learning Practice

  • 9.1

    Introduction

    This section introduces the concepts of GCD properties and Bezout's theorem, emphasizing their relevance in number theory and algorithms.

    Learning Practice

  • 9.2

    Bezout’s Theorem

    Bezout's theorem establishes the relationship between the GCD of two integers and their integer linear combinations.

    Learning Practice

  • 9.3

    Proof Of Bezout’s Theorem

    Bezout's Theorem states that the greatest common divisor (GCD) of two integers can be expressed as a linear combination of those integers.

    Learning Practice

  • 9.4

    Extended Euclid’s Algorithm

    The Extended Euclidean Algorithm provides a method to find the greatest common divisor (GCD) of two integers and expresses it as a linear combination of the two integers.

    Learning Practice

  • 9.5

    Multiplicative Inverse Modulo N

    This section covers the concept of modular multiplicative inverses, including the conditions under which they exist and how to find them using the extended Euclidean algorithm.

    Learning Practice

  • 9.6

    Existence Of Multiplicative Inverse

    This section discusses the properties of the multiplicative inverse in modular arithmetic, including conditions under which it exists based on the GCD of two numbers.

    Learning Practice

  • 9.7

    Summary

    This section covers properties of the GCD, Bezout's theorem, and the extended Euclidean algorithm.

    Learning Practice

References

ch58.pdf

Class Notes

Memorization

What we have learnt

  • Bezout's theorem states tha...
  • The extended Euclidean algo...
  • A multiplicative inverse of...

Final Test

Revision Tests