## Introduction to Number Theory

Number theory, mathematics branch concerned with the properties of positive integers (1, 2, 3, ...). Sometimes referred to as "higher arithmetic," it is one of the oldest and most natural mathematical pursuits. Number theory helps to study the relationship between different types of numbers.

There were no direct applications to the real world of number theory in the past. The emergence of digital computers and digital media showed that number theory could provide surprising answers to real-world problems. Simultaneously, computer technology improvements have made it possible for number theorists to make remarkable advances in factoring large numbers, determining primes, etc.

## What is Number Theory

Numbers are theoretically classified into different types, such as natural numbers, whole numbers, complex numbers, and so on. The subclassifications of the natural number are given below:

Odd Numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25. . .

Even Numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 . . .

Square Numbers: 4, 9, 16, 25, 36, 49, 64, 81,100 . . .

Cube Numbers: 8, 27, 64, 125, 216, 343, 512 . . .

Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, . . .

## Number Theory Topics

Even Numbers: Integers that can be divided exactly by 2 are called even numbers. Examples of even numbers are 2, 4, 6, 8, 10, 12, 14, 16. These are even numbers as these numbers can easily be divided by 2.

Odd Numbers: Odd numbers are described as any number that is not divisible by 2. Examples of odd numbers 1, 3, 5, 7, 9, 11. 1 is the first positive odd number on the number line.

Square Numbers: The resultant is called a 'Square Number' when a number is multiplied by itself. Square numbers can also be called perfect square numbers. Examples of square numbers are 1, 4, 9, 16, 25, etc.

Cube Numbers: The result of multiplying a whole number by itself twice. Examples of cube numbers are 1, 8, 27, 6, etc. 1, 8, 27, 25, are the cube of 1, 2, 3, 4 respectively.

Prime Numbers: The positive integers with only 2 factors, the number 1 and the integer itself, are prime numbers. For example, factors of 8 are 1,2,4 and 8, which are four factors in total. But factors of 5 are only 1 and 5, totally two. Therefore, 5 is a prime number, but 8 is not, but it is a composite number instead. But always remember that 1 is neither composite nor prime.

Composite Numbers: The numbers that have more than 2 factors are composite numbers, unlike prime numbers that have only 2 factors, i.e. 1 and the number itself. Such numbers are called composite numbers. As they can be divided by more than two numbers, all-natural numbers that are not prime numbers are composite numbers. For instance, 6 is composite, since it is divisible by 1, 2, 3, and 6.

Fibonacci Numbers: The Fibonacci number is a number sequence where a number is the addition of the last two numbers, starting with 0, and 1.

Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

X_{n} = X_{n-1} + X_{n-2}

## Applications of Number Theory

Number theory is used to find out whether a given integer m divides the integer n by some of the significant divisibility tests. Number theory has many applications in both mathematics and practical applications, such as:

Barcodes.

Creation of modular designs.

Method of Memory Management.

Device authentication.

Security system like banking securities.

Websites for E-commerce.

Theory of Coding.

1. Show that the Number Itself is the Greatest Factor of Itself.

Answer: Let’s take the number 20.

The factors of 20 are

1 × 20 =20

2 × 10 = 20

4 × 5 = 20

5 × 4 = 20

10 × 2 =20

20 × 1 =20

The factors of 20 are 1, 2, 4, 5, 10, and 20.

We can say from this that 20 is the greatest factor of the number 20. Hence proved.

2. Find the Common Factors of 10 and 16.

Ans: Factors of 10 are:

2 × 5 =10

1 × 10 = 10

Factors of 10: 1, 2, 5 and 10

Factors of 16 are:

4 × 4 = 16

1 × 16 = 16

2 × 8 = 16

Factors of 16: 1, 2, 4, 8 and 16.

Common factors of 10 and 16 are 1 and 2.