How to find if a number is prime? We get questions on this
topic in some competitive exams. Deciding whether a number is prime number or
not is not a simple task if it is bigger number. So, some techniques are needed
to tell if the number is prime. This is what you are looking for, then you are
in right place. Because I have covered everything you need to know about Prime
numbers.

### Contents :

####
1. Prime Number Definition
2. Prime Factorization – Finding Prime Factors
3. Finding all prime numbers between 1 and
100 in a simple way
4. How to check if a number is Prime
5. Some facts about prime numbers
6. prime numbers upto 1000

### Prime Number Definition :

A Number which is divisible by 1 and the number itself is called as PRIME NUMBER.

Prime number must be a positive integer and it should be
greater than 1.

**Ex :**

17 is divisible by 1 and 17 only, So, it is prime number.

15 is divisible by 1,3,5,15. So, it is not prime number.

### Prime Factorization – Finding Prime Factors :

Factors of any number should be prime numbers in prime factorization.

Factors are numbers which are multiplied to get original
number.

2 * 3 = 6 => 2, 3 are factors of 6. They
are also prime factors of 6.

54 = 9 * 6 => 9,6 are factors of 54,
but they are not prime factors.

#### Factor Tree :

Factor Tree can be used to get prime
factors of a number.

2 * 3 * 3 * 3 = 54 => this is how we do prime
factorization.

2,3 are prime factors of 54.

### Finding all prime numbers between 1 and 100 in a simple way :

25 prime numbers are there in between 1 and 100.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97…………

- First arrange numbers in a table like shown in figure.
- Enter 6 numbers in each row until the last number (in this it is 100) reaches.
- Start with 2 which is greater than 1.
- Round off number 2 and strike off entire column until the end.
- Similarly strike off 4
^{th}column and 6^{th}column as they are divisible by 2. - Now round off next number 3 and strike off entire column until end.
- 4 is gone.
- Now round off next number 5 and strike off numbers in inclined fashion as shown in figure. When a striking off ends in some row, start again striking off with number in another end which is divisible by 5 in a row parallel to previous strike off line as shown in figure.
- 6 is gone.
- Now round off number 7 and strike off numbers as we did in case of number 5.
- 8,9,10 are gone.
- Stop at this point. Wonder why do we need to stop??
- It’s a trick that we have to do this procedure until we reach square root of bigger number.
- This technique was developed in 3
^{rd}century B.C.

### How To check if a Number is Prime :

Divide the given number with every number below it except 1 and the number itself.

Check whether the given number is divisible by any of the numbers below it.

If it is not divisible by any number then it is prime number.

"Divisible by" means when you divide the whole number with another whole number result should be whole number with remainder zero.

Actually you don’t need to divide with all numbers.

First check whether it is divisible by 2 or not.

Next check with 3, 5, 7, 11,13…..

Check divisibility only with prime numbers.

Here also you don’t need to divide with all prime numbers below it.

Let p be the given number.

Find a number n greater than \(~\sqrt{p}\)

Find prime numbers below n and divide p
with only those prime numbers below n.

Actually, we used the same procedure in the above figure of finding all prime numbers upto 100.

Actually, we used the same procedure in the above figure of finding all prime numbers upto 100.

**Ex : 1 - Number is 149**

Let us find whether 149 is prime or not.

\[\sqrt{149}= 12.20 < 13\]

prime numbers below 13 are 2,3,5,7,11….

\[\sqrt{149}= 12.20 < 13\]

prime numbers below 13 are 2,3,5,7,11….

149 is not exactly divisible by
2, because it dont has even number at end.

149 is not exactly divisible by 3, because the sum of numbers 1+4+9= 14 is not divisible by 3.

149 is not exactly divisible by 5, because it dont has 0 or 5 at end.

149 is not exactly divisible by 7,11.

So, 149 is prime number…

149 is not exactly divisible by 3, because the sum of numbers 1+4+9= 14 is not divisible by 3.

149 is not exactly divisible by 5, because it dont has 0 or 5 at end.

149 is not exactly divisible by 7,11.

So, 149 is prime number…

**Ex: 2 - number is 631**

Let us find whether 631 is prime or not.

\[\sqrt{631}= 25.11 < 26\]

prime numbers below 26 are 2,3,5,7,11,13,17,19,23….

\[\sqrt{631}= 25.11 < 26\]

prime numbers below 26 are 2,3,5,7,11,13,17,19,23….

631 is not exactly divisible by 2,3,5,7,11,13,17,19,23…

So, 631 is prime number…

For easy divisibility checking, you should know the divisibility rules.

###
Some Facts on Prime Numbers:

Here is the list of prime numbers upto 1000 for your reference.

Thats all about Prime Numbers.

Read again about Basic Number theory

check about blood relations tricks

So, 631 is prime number…

For easy divisibility checking, you should know the divisibility rules.

###
Some Facts on Prime Numbers: ** **

**Co Primes:**

H.C.F. of two numbers in one set should be 1.

Ex: (2,3),(4,5),(6,7),(11,15)

If a number N is divisible by two numbers a and b, where a, b are co primes, then N is divisible by ab.

**Twin Primes:**

Difference between prime numbers in one set should be 2.

Ex: (3,5),(5,7),(11,13),(17,19),(31,33)

If a Number M is a prime number and N is a next prime number, the average difference between M and N is “ ln(M) ” [natural logarithm of M]

- 2 is the only one prime even number.
- Numbers greater than 1 and which are not prime numbers are composite numbers.
- Some probabilistic methods are available for checking big prime numbers.

### Prime Numbers upto 1000 :

Here is the list of prime numbers upto 1000 for your reference.

Thats all about Prime Numbers.

Read again about Basic Number theory

check about blood relations tricks