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IB Mathematics Extended Essay Exemplar - Grade A £4.99   Add to cart

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IB Mathematics Extended Essay Exemplar - Grade A

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In 2020, I received 45 points in IB, including 3 core points. This Extended Essay allowed me to achieve that grade. Understanding its style, structure and presentation will help boost your EE mark considerably. The essay's title is, 'What is modular arithmetic, and how can it be applied to solve...

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  • July 30, 2022
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  • 2020/2021
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Extended Essay

Mathematics




What is modular arithmetic, and how can
it be applied to solve problems in the field
of number theory?

An essay on linear congruences and their uses




Word count: [3989]

,Contents


1 Introduction


2 Modular Arithmetic

2.1 Introduction

2.2 Operational Properties

3 Significant Theorems

3.1 The Chinese Remainder Theorem

3.2 Fermat’s Little Theorem


4 Divisibility Tests

5 Number Theory Problems


6 Conclusion

7 Appendices

8 Bibliography




2

,1 Introduction

In this essay, I will answer the research question, ‘What is modular arithmetic,

and how can it be applied to solve problems in the field of number theory?’

Initially, the basic concept of modular arithmetic will be introduced, before

establishing its foundational properties. Subsequently, I will set out two

significant theorems that are central to the topic and to the 5 original solutions

to number theory problems that I will present in the last section of my essay, the

majority of which are posed by the United Kingdom Mathematics Trust (UKMT).



A variety of sources were used in my research, including the online resources

Encyclopaedia Britannica [1] and Brilliant.org [2] and the books Concepts of

Modern Mathematics [3] and Concrete Mathematics [4].



Modular arithmetic is a topic that has tremendous significance and applicability

in pure mathematics, principally in number theory, along with practical uses in

cryptography and checksum calculations. Having been a keen participant in

UKMT Mathematics Challenges since I was 13, I find modular arithmetic

particularly compelling as it can be utilised in order to produce succinct and

elegant solutions to the types of questions I have enjoyed doing for several years.



The aim of this essay is two-fold: to outline what modular arithmetic is and to

clearly demonstrate its usefulness in solving number theory problems, as

evidenced by my own original solutions.




3

, 2 Modular Arithmetic

2.1 Introduction


Established by Carl Friedrich Gauss in his 1801 book ‘Disquisitiones

Arithmeticae’ (Arithmetical Investigations) [3], modular arithmetic is a system of

arithmetic solely concerned with the remainders of integers when they are

divided by a chosen value, called the modulus. Gauss introduced the following

notation:



Let a, b, r ∈ ℤ and n ∈ ℤ+.

The remainder, r, when a is divided by the modulus, n, is denoted by:

𝑎 ≡ 𝑟 (mod 𝑛).



If a and b have the same remainder when divided by n, they are congruent

modulo n. This is written as:

𝑎 ≡ 𝑏 (mod 𝑛)

or

𝑎 mod 𝑛 = 𝑏 mod 𝑛 .



For example,

16 ≡ 9 (mod 7)

because 16 and 9 both leave a remainder of 2 when divided by 7.




4

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