Prime, Coprime and Composites

Prime Numbers
A natural number larger than unity is a prime number if it  does not have other divisors except for itself and unity.
Note:-Unity i e,1 is not a prime number.

 Properties Of Prime Numbers:
->The lowest prime number is 2.
->2 is also the only even prime number.
->The lowest odd prime number is 3.

–> 2 and 5 are the only primes that end in 2 or 5.  

Remember :  

The remainder when a prime number p>=5 s divided by 6 is 1 or 5.
However, if a number on being divided by 6 gives a remainder 1 or 5 need not be prime. 

The remainder of the division of the square of a prime number p>=5 divided by 12 is 1. 
 

Remember 

 >If a and b are any 2 odd primes then a²-b² is composite. Also a²+b² is composite.

Remember:
If x is a prime number then number of the form (a^x – a) is always divisible by x.



Remember:
A composite number n which is not a perfect square has one factor
greater than √ n one less than √ n , product of these factors will give
the number.



Remember:
If p is a prime number such that  the

p^2 = 6K +1  where p>3
p^2 = 24k + 1  where p >= 5
In short p^2 - 1 is divided by 6 and 24


One just needs to remember the above rule. Still for deriving that formula one can Google to get it.

->Goldback's Conjecture
- if n is an even number greater than 2, then there are always 2 prime numbers whose sum is n
 

Process to Check A Number s Prime or not:
Take the square root of the number.
Round of the square root to the next highest integer call this number as Z.
Check for divisibility of the number N by all prime numbers below Z. 
If there is no numbers below the value of Z which divides N then the number will be prime.

Example 239 is prime or not?
√239 lies between 15 or 16.Hence take the value of Z=16.
Prime numbers less than 16 are 2,3,5,7,11 and 13.
239 is not divisible by any of these. Hence we can conclude that 239 
is a prime number.


Composite Numbers
The numbers which are not prime are known as composite numbers.

Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.

4 is first composite number.

Co-Primes (Relatively Prime Numbers):
Two numbers a an b are said to be co-primes,if their H.C.F is 1.
Example (2,3),(4,5),(7,9),(8,11).....

For instance (9,16)
Factors of 9 -> 1, 3, 9
Factors of 16 -> 1, 2, 4, 8, 16

Remember:-
1. One is co-prime with any other number. (1,2) , (1,3) …
2. Two Consecutive numbers are always co-prime.
3. A prime number is co-prime with all numbers except its multiple.
4. Two prime numbers are always co-prime. 

Twin Prime Numbers
Pair of prime numbers with the difference of 2 is called twin prime number.

Eg. (3,5) , (5,7) , (11,13) and go on.