## Integers and its properties

### Subsets of Z (set of integers)

- Odd numbers: {......-5, -3, -1, 1, 3, 5......}
- Even numbers: {......-4, -2, 0, 2, 4, 6, 8......}
- Negative integers: {......-4, -3, -2, -1}
- Non-negative integers (Whole Numbers): {0, 1, 2, 3, 4......}
- Positive integers (Natural Numbers): {1, 2, 3, 4, 5......}
- Prime Numbers: {2, 3, 5, 7, 11, 13, 17......}
- Composite numbers: {4, 6, 8, 9, 10, 12, 14, 15......}

### Properties of odd and even numbers

#### Property I: Addition

- Sum of odd number of odd numbers is odd. Eg: 1 + 3 + 5 = 9.
- Sum of even number of odd numbers is even. Eg: 3+ 11 + 7 + 5 = 26.
- Sum of any number of even numbers is even.
- In a certain summation of odd and even numbers, then the sum is:
- Odd: when there is odd number of odd numbers in the selected group of integers
- Even: when there is even number of odd numbers in the selected group of integers

#### Property II: Subtraction

What are the properties considered in the addition of integers, the same properties are applicable in the subtraction in integers.

- If `a + b` is odd for any two integers a and b, then; `a - b`, `-a + b`, `-a - b` are all odd.
- If `a + b` is even, then `a - b`, `-a + b`, `-a - b` are all even.

#### Property III: Multiplication

If a group of integers consist at least one even number then the product of the given integers should be even, otherwise the product become odd (ie. all integers are odd).

#### Property IV: Division

Consider a rational number `x/y`, where x and y are two integers and `y != 0`.

##### Case I: Numerator is even and denominator is odd:

In a division in the form `x/y`, if the numerator is an even multiple of the odd denominator y, then the result of `x/y` is always an even number. Example: `36/9 = 4` (it is even).

##### Case II: Both the numerator and denominator are even:

In `x/y`, where x and y are integers and x is divisible by y, then it is not possible to make a conclusion of the nature of the value of `x/y`. Example: `10/5 = 2` (it is even)

##### Case III: Dividing an odd number by an even number:

In `x/y` , if x is odd and y is even, then x can't be any multiple of y. Hence the value of `x/y` in this situation can't be an integer.

##### Case IV: Divide an odd number by another odd number:

In the division `x/y`, both x and y are odd numbers and x is divisible by y, then the value of `x/y` is always odd Example: `15/3 = 5` (it is odd)

#### Some interesting properties of odd numbers

- Cube of all natural numbers greater than '1' can be expressed as the sum of a certain number of consecutive odd numbers.

`n^3 = [(n - 1). n + 1] + [(n - 1). n + 3] + [(n - 1). n + 5] + ... + [(n-1). n + (2n - 1)]`

Examples:- `23 = 3 + 5`
- `33 = 7 + 9 + 11`
- `43 = 13 + 15 + 17 + 19`

- All odd natural numbers except '1' can be one of the three values of a Pythagorean triplet.
- 32 + 42 = 52
- 52 + 122 = 132
- 72 + 242 = 132
- 92 + 402 = 412
- 112 + 602 = 612
- 132 - 122 = 52
- 152 + 82 = 172
- 172 - 152 = 82

## Prime and Composite numbers

### Prime number

A natural number which doesn't have any factor other than 1 and itself is a prime number. Example: 2, 3, 5, 7, 11, 13, 17 etc.

#### Properties of prime numbers

- 2 is the least prime number
- 2 is the one and only one even prime.
- All primes greater than 3 can be expressed in either of the form `6k - 1` or `6k + 1`, where `k` is any natural number, but all the numbers which are in the above mentioned forms are not necessarily prime always. Eg when k = 4, `6k + 1 = 25`, it is not a prime.
**Goldbach's Conjecture:**Every even natural number greater than 2 can be expressed as the sum of two primes ( not necessarily distinct). Eg: `4 = 2 + 2`, `6 = 3 + 3`, `8 = 3 + 5`, `10 = 3 + 7` and so on.**Relatively prime / co-prime numbers:**If two natural numbers doesn't have a common factor other than 1 are called relatively prime/co-prime numbers. Eg. 8 and 9 are relatively prime. 4 and 5 are co-prime.- 1 is relatively prime to all the other natural numbers.
**Twin Primes:**If any two consecutive odd numbers are prime, ten they are called twin primes. Eg. (3,5), (5,7), (11,13), (17,19), etc.**Prime triplet:**If three consecutive odd numbers are prime, then they are called prime triplets. There is only one prime triplet and that is (3,5,7).**Fermatt's theorem:**If 'a' and 'b' are any two prime numbers, then `a^b - a` is always divisible by b. Eg: `2^3 - 2 = 6` is divisible by 3.

`3^5 - 3 = 240` is divisible by 5**Wilson's theorem:**For any prime 'p', `(p - 1)! + 1` is divisible by p. Eg: Let p = 5, `(p - 1)! + 1 = 4! + 1 = 25` is divisible by 5.

Let p = 11, `(p - 1)! + 1 = 10! + 1 = 3628800 + 1 = 3628801` is divisible by 11.**Progression of prime numbers:**Some set of prime numbers are forming an Arithmetic Progression. Eg: 11,17,23,29

199,409,619,829, 1039, 1249, 1459, 1669, 1879, 2089.**How to identify a prime number?**Consider 173. For checking 173 a prime or not, consider the square of any prime number which is immediately less tan 173. Here `13^2 = 169` is immediately less than 173.

Check the divisibility of all the prime numbers up to 13 with 173. None of the prime numbers 2, 3, 5,7, 11 and 13 is not a factor of 173. Therefore 173 is a prime number.

Example: Is 391 a prime number?

Square of a prime number, which is immediately less than 391 is `19^2 = 361`. While checking the divisibility of all primes up to 19, we will get 17 is a factor of 391 (Ie. `391 = 17 xx 23`). Therefore 391 is not a prime number.- In the set of natural numbers from 1 to 50 there are exactly 15 prime numbers and from 51 to 100, there are 10 prime numbers.
- For any prime `p > 3`, `p^2 - 1` is a multiple of 24.

#### Distribution of prime numbers in the first thousand natural numbers

Range of natural numbers | Number of prime numbers |
---|---|

1 to 100 | 25 |

101 to 200 | 21 |

201 to 300 | 16 |

301 to 400 | 16 |

401 to 500 | 17 |

501 to 600 | 14 |

601 to 700 | 16 |

701 to 800 | 14 |

801 to 900 | 15 |

901 to 1000 | 14 |

### Composite number

If a natural number has at least one factor other than 1 and itself is called a composite number. Eg: 4, 6, 8, 9, 10, 12, 14, 15 etc.

#### Properties of composite numbers

- 4 is the least composite number.
**Wilson's theorem:**For any composite number `c > 4`, `(c - 1)!` is divisible by c.

Eg: Let c = 6

`(c - 1)! = (6 - 1)! = 5! = 24` is divisible by 6.

Let c = 12

`(c - 1)! = 11! = 39916800` is divisible by 12.