Euclid around 300 BC stated that Prime numbers are more than any assigned quantity of prime numbers.

According to Euclid there are infinite prime numbers.

Prime numbers are like siblings, although members of the same big family resembling each other, but not quite alike.

Which is the oddest of all prime numbers? It is 2 mainly because it is the only even prime number.

The number 1 has only one positive divisor and that is one itself.
Every number (a), where a>1, would have at least two positive divisors, ‘1’ and ‘a’
A number "a > 1" is called a prime number or we simply call it prime if it has only two positive divisors.

## What is a Prime Number?

A prime number is a whole number larger than the number 1 and one that can be equally divided only by itself and 1.
All prime numbers beyond 2 are considered odd because all even numbers can be divided evenly by 1 themselves and by 2.
So even numbers larger than 2 won’t fit into the definition of a prime number.
Since prime numbers are not formed by multiplying any two or more numbers they cannot be divided by any number.

Prime number definition: A prime number is a positive integer which has only two factors, which can be 1 and the number itself.

For example, lets consider the number 2.

2 = 2 × 1

So, 2 has factors 2 and 1. It means it has only two numbers as factors. Hence it is a prime number.

Now, lets consider the number 6.

6 = 3 × 2

= 3 × 1 × 2 × 1

= 3 × 2 × 1

6 has three factors 3, 2 and 1. So it is not a prime number.

Such numbers are called composite numbers.

The first few prime numbers are: 2 , 3 , 5 , 7 . ...............

There are various methods to find out what are prime number, they are:

1. Sieve of Eratosthenes

This provides a method to test a number to see if it is prime.

The method is as follows:
Write the list of numbers starting from 2 and going up to any limit, say 15 .

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

The first prime number is 2. Circle 2 and cross out all the multiples of 2

The next unmarked number is 3 , so circle it and cross out all of its multiple.

Next, its 5 . Circle it and cross out all of its multiple.

Similarly we do it up to 19.

Taking all the circled numbers,

2 , 3 , 5 , 7 , 11 and 13

We get the list of first prime numbers, up to 15 .

2. Another method for determining whether a number is prime is to divide the number by all primes less than or equal to the square root of that number. If any one primes of the divisions forms an integer, then the original number is not a prime. Otherwise, it is a prime.

## Properties of Prime Numbers

1 is not a prime or composite number.
Hence, prime number can be redefined as:
An integer greater than one with only positive divisors as one and itself.

2 is the only even number in the set of prime numbers.
Hence, the term odd prime number means any prime number greater than two.

• The number of prime number is infinite.
• The property of being prime is called primarily.
• It can be seen that all prime numbers except 2 and 5 end in 1 , 3 , 7 or 9 because multiples of 2 ends in 0 , 2 , 4 , 6 or 8 and multiples of 5 ends in 0 or 5.
• If p is a prime number and p divides the product ab of integers, then p divides a or p divides b.

## Prime Numbers upto 100

Prime Number List:

List of prime numbers up to 100 in the prime numbers chart

a) Cross 1 out.
b) Then start from 2, circle 2 and cross out every multiple of 2 from the prime table.
c) Start from 3, circle 3 and cross out every multiple of 3 from the prime table.
d) Start from 5, circle 5 and cross out every multiple of 5 from the prime table.
e) Start from 7, circle 7 and cross out every multiple of 7 from the prime table.
f) Repeat the same with all the prime numbers which you know already. The numbers which are crossed are not prime numbers, because those numbers are multiples of other numbers. The number which are circled are prime numbers.

## List of Prime Numbers

There are an infinite number of prime numbers and it is a difficult task to memorize or recognize them, but it is always a good thing to remember the prime number chart comprising of smaller prime numbers.

The list of prime numbers up to 200 is provided below. The list of prime numbers for the first 100 (0 -100) numbers include 25 prime numbers and thereafter another 25 prime numbers (101-200) till 200.

Here is the list of the first 46 prime numbers

## Prime Number Chart

The prime number chart given below includes all the numbers within one and thousand (1-1000). In this prime number chart, the prime numbers that differ with one another by 2 are called twin primes. The chart will have all prime numbers within 1-1000 and will have lots of twin primes as well.

## Is 1 a prime number?

By rules and definition, the number 1(one) is not a prime number.
By definition a prime number can be divided evenly only by itself and 1, but in this case 1 is itself and so won’t fit into the definition.
The number 1, (one) is considered as just a generator of all other even and odd numbers.

## Is 2 a prime number?

By definition rule 2 matches the characterization of a perfect prime number.
By convention it cannot be divided by any other number other than 1 and itself and at the same time it is also considered as an even number.
This number is henceforth considered as an ‘even’ prime number and all the numbers beyond 2 are all ‘odd’ prime numbers.

## Largest Prime Number

The list of prime numbers as such will never end.
There is no largest prime number as such and over the years many mathematicians have been fascinated by prime numbers and they are still continued to be.
The largest prime number known as of date requires around four million digits to write about and the search is on for finding even larger prime numbers.
If we assume that 'n' is the largest prime number then we can construct a prime number larger than n > N by multiplying all the prime numbers up-to 'N' and adding 1.
This assumption is considered to be in contradicting to the fact that there is a prime number larger than the largest prime number, and so we can safely reject the assumption.

The largest prime number is the largest integer which is also know as largest known primes.

## Relatively Prime

If any two numbers have no common factor other than 1 then they are assumed to be relatively prime. If two numbers are relatively prime, then their greatest common factor is 1 and in case the two numbers are not relatively prime then their greatest common factor is the largest number that will divide both numbers equally.

For example we have two numbers 16 and 15 where the factors for 15 are 3, 5, and 15. The factors for 16 are 2, 4, 8, and 16. We can see that both 15 and 16 are composite numbers yet they are relatively prime. In case we are taking numbers a, b, c, d…as relatively prime numbers then the least common multiple of the two relatively prime numbers ‘a’ and ‘b’ is their product ‘ab’.

We could also say that if a, b, c, d… are relatively prime numbers, then each number divisible by all of them is also divisible by their product.

Example: Check whether 12 and 18 are relatively prime?
The factors of 12 are as follows: 2, 3, 4, 6, and 12
The factors of 18 are as follows: 2, 3, 6, 9, and 18
As per the rule and definition we could see that both the numbers have 2, 3, 6 as common factors.
These are not relatively prime number.

Two integers are said to be relatively prime if they have no common positive factors except 1.
It means if we consider any two integers, they will share only one factor as common, which will be one.

Relatively prime numbers are sometimes called strangers or co-prime numbers.

### Relatively Prime Examples

Below are the examples based on relatively prime integers

Example 1:

The number 3 and 5 are relatively prime to each other.
since 3 = 3 × 1 and 5 = 5 × 1 , so it has only one as common factor.

Now, consider 3 and 6.
3 = 3 × 1 and 6 = 3 × 2 × 1
They both have 3 as common other than one. So, 3 and 6 are not relatively prime.

All prime numbers will be relatively prime.
For example, consider 7 and 11 .
They are prime numbers. So,
7 = 7 × 1 and 11 = 11 × 1
So they don't have any factors common than one.
So 7 and 11 are relatively prime.

If we consider the greatest common divisor,two integers m and n are relatively prime if there greatest common divisor is 1 .

So, the two relatively prime numbers need not be prime.

Example 2:

Consider 16 and 21
Factors of 16 are 1 , 2 , 4 , 8 , and 16
and factors of 21 are 1 , 3 , 7 and 21 .

They don't have any factors other than 1 common but they are not prime.
So all relatively prime numbers are not prime.

The method for calculating the number of relatively prime numbers less than a given
number is as follows:

Step 1. Find the prime factorization of the number and write it in exponential form.

Step 2. Taking each term separately,

a. Subtract 1 from the base.

b. Subtract 1 from the exponent of the base and evaluate the expression.

Step3. Multiply all the numbers together.

Example 3:

How many numbers less than 20 are relatively prime to 20 ?

Solution:
The prime factorization of 20 is: 22 × 51.

Taking 22 first, we get: 2 - 1 = 1 and 2(2 - 1) = 2.

Taking 51 we get: 5 - 1 = 4 and 5(1-1) = 1.

Multiplying all of them together = 1 × 2 × 4 × 1 = 8.

The numbers which are relatively prime are 1 , 3 , 7 , 9 , 11 , 13 , 17 , and 19 . So indeed there are 8 .

Another way of defining relatively prime number is that "Two positive integers are said to be "relatively prime" if 1 is the only number that divides both of them evenly."

All prime numbers are co-prime to each other.

Number 1 is co-prime to every integer.

Relatively prime numbers can also have only -1 as common factor.

Hence, any two integers x and y are said to be relatively prime if they have only common factor 1 or - 1.