# NUMBER THEORY:

* Number theory* is the study of positive numbers. These positive integers are a set of

**. Sometimes we add zero in a set of positive numbers in the**

*natural numbers*

*number***then that set is called a set of whole numbers but when we skin the zero then that is called a set of natural numbers. This shows us that it zero is added in**

*theory***then they become whole numbers and set of natural numbers become equal to set of whole numbers ( N = W). As N belongs to “natural numbers” and W belongs to “whole numbers”.**

*natural numbers** Number theory* is the concern of numbers and especially with the

*s and this is further divided into few branches of numbers like odd numbers, even numbers, Fibonacci numbers, perfects, squares, 1(modulo 4), 3(modulo 4), etc.*

**set of natural number**Let me give you some brief description of all these numbers too.

### Odd: a set of numbers do not divisible by 2 like { 1,3,5,7,…}

Example: ½=0.5

### Even: set of numbers that are divided by 2 like: { 2,4,6,8,…}

Example : 2/ 2 = 1 , 4/2 = 2…

### Square: a set of numbers that makes the square like 1 square is 1, 2 square is 4 then the square of 3 is 9, etc. so likewise it would be {1, 4, 9,…}

1 square = 1

2 square = 4

3 square = 9…

### Cube: a set of those numbers which make cube like a cube of 1 is 1 and cube of 2 is 8 etc. so it would be like {1, 8, 27, 64,…}

1 cube is equal to 1

2 cube is equal to 8

3 cube is equal to 27…

### Prime: set of numbers other than one which is just divided by one and on its self. { 2, 3, 5, …}

Example: 2 is divided by one and 2 just.

3 is divided by one and 3 just.

5 is divided by one and 5 just.

### Composite: a set of numbers other than 1 that are not prime and that are divisible by more than 2 numbers. Like { 4 , 6, 8, 9 , 10 , 12, …}

Example: 4 is divided by one, 2 and 4.

6 is divided by one, 2, 3, 6. …

### 1(modulo 4): set of those numbers which when divided by 4 gives the remainder 1. Like { 1, 5, 9 , 13 …}

Divide 1 with 4 we will get 1

5 divides by 4 we will get 1…

Similarly for 3(modulo 4) which gives 3 remainders when divided by 4. Like { 3, 7, 11, 15,…}

### Perfect: a set of those numbers which gives again that original number when we add all of its divisors. Like : { 6 , 28 , 496,…}

Example: divisors of 6 are 1, 2, 3 now if we add them we will again get 6 like 1+2+3=6…

### Fibonacci: a set of all those numbers which gives the next number by adding last two numbers, like { 1, 1, 2, 3, 5, 8, 13, 21, ..}

Example: 1+1=2

1+2=3

5+8=13…

And so on.

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