Discrete Mathematics - Vol 3 | 11. Uniqueness Proof of the CRT by Abraham | Learn Smarter
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11. Uniqueness Proof of the CRT

The discussion focuses on the uniqueness proof of the Chinese Remainder Theorem (CRT) and highlights important properties such as Euclid's Lemma and basic properties of divisibility. Emphasizing the proof strategy involves demonstrating that if two numbers yield the same results under a set of linear congruences, they must be identical within a specified range. Real-world applications of the CRT, particularly in cryptography and arithmetic with large values, are emphasized, showcasing its practical significance.

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Sections

  • 11.1

    Discrete Mathematics

    This section explores the uniqueness of solutions to systems of linear congruences as guided by the Chinese Remainder Theorem (CRT).

    Learning Practice

  • 11.2

    Uniqueness Proof Of The Crt

    This section focuses on proving the uniqueness of solutions for a system of linear congruences using the Chinese Remainder Theorem (CRT).

    Learning Practice

  • 11.2.1

    Properties Of Divisibility

    This section explores fundamental properties of divisibility, including results from Bezout's theorem and Euclid's lemma, crucial in the context of linear congruences.

    Learning Practice

  • 11.2.2

    Euclid’s Lemma

    Euclid's Lemma asserts that if a prime divides a product of integers, it must divide at least one of those integers.

    Learning Practice

  • 11.2.3

    Uniqueness Proof Part For The Chinese Remainder Theorem

    This section focuses on proving the uniqueness of solutions for the Chinese Remainder Theorem.

    Learning Practice

  • 11.2.4

    Helping Lemma

    This section discusses the uniqueness of solutions in linear congruences and the role of the Helping Lemma in the Chinese Remainder Theorem (CRT).

    Learning Practice

  • 11.2.5

    Example Of Chinese Remainder Theorem

    This section discusses the Chinese Remainder Theorem, focusing on the uniqueness of solutions to systems of linear congruences.

    Learning Practice

  • 11.2.6

    Application Of Chinese Remainder Theorem

    The section details the application of the Chinese Remainder Theorem (CRT), emphasizing the uniqueness of solutions in linear congruences.

    Learning Practice

References

ch60.pdf

Class Notes

Memorization

What we have learnt

  • The Chinese Remainder Theor...
  • Euclid's Lemma is a key pro...
  • The theorem is applicable t...

Final Test

Revision Tests