PRIME AND COMPOSITE NUMBERS

Prime And Composite Numbers
Observe the number of factors of some numbers arranged in the following table

We can see that
i)    The number ‘1’ has one and only one factor  (i.e., itself)
ii)    There are numbers having exactly two factors 1 and the number itself. we can see that such numbers are 2, 3, 5, 7, 11 etc.
    These numbers are said to be prime numbers

Definition:- 
    The number which has only factors 1 and itself is called prime number.
Note :- ‘2’ is only even prime number

iii)     There are numbers which have more than two factors like 4, 6, 8, 9, 10 etc and so on are called composite numbers

Definition :-
     Numbers having more than two factors are called composite numbers.
Note :-  ‘1’ is neither prime nor composite number.
    
By using the method of sieve of eratosthenese easily we can verify the whether the given number is prime or not

SIEVE OF ERATOSTHENES:-
     

Step -I-  Cross  out ‘1’ because it is not a prime number
Step -II- Encircle 2, cross out all the multiples of 2, other than 2 itself i.e., 4, 6, 8 and so on
Step III - We will find that the next uncrossed number is 3, encircle 3 and cross out all the multiples of 3, other than 3 itself .
Step IV- The next uncrossed number is 5 encircle 5 and cross all the multiples of 5 other than 5 itself.
Step V- Continue this process till all the numbers in the list are either encircled or crossedout.
  
Now, if we listout all the encircled numbers are prime numbers. All the  crossed out numbers. other then ‘1’ are composite numbers.  This method is called the sieve of eratosthenes.

Twin Primes :- If the difference of two prime numbers is ‘2’ then they are called Twin primes.
    Ex:- (3, 5), (5, 7), (11, 13)............etc

Co-primes or relative primes :- Two numbers are said to be co-primes or relatively prime numbers if their H.C.F is ‘1’
    Ex:- (3, 5), (10, 11), (15, 16)............ etc are co-primes.

Note: 1) The co-primes need not necessarily be prime themselves.
          2) A pair of co-primes may consist of 
        a) both primes eg :3, 5        
        b)  one  prime and one composite eg: 7, 6
        c) both composite eg:-8, 15
        d) The co-primes need not necessarily be prime themselves 
        e) If two numbers are not co-primes then they must have a common factors other than ‘1’

Some more Important points
1.    There are only two primes which are consecutive integers.  Those are 2 and 3.
2.    The primes 3, 5 and 7 are called prime triplet
3.    If a and b are prime numbers, then their product ab will have only a, b and ab as factors
    Ex : a = 3, b= 5, ab = 15 
    factors of 15 are 1, 3, 5, 15
4.    Set of prime numbers is infinite.

EVEN NUMBERS:- 
    All numbers divisible by ‘2’ are called even numbers 2, 4, 6, 8 ......

ODD NUMBERS:- 
    Numbers which are not divisible by ‘2’ are called odd numbers 1, 3, 5........
    General form of even and odd numbers
1.    General form of even number can be written as 
    General form of odd numbers can be written as 
2.    Ex:- E={0,2,4...},O={1,3,5…….} , 
    
Some facts about even and odd numbers
1.    The sum or product of any number of even numbers is even
    Ex:- 4 + 2 = 6 and 
2.    The difference of two even numbers is even 
   Ex:- 4 - 2 = 2
3.    The sum of odd numbers depends on the number of numbers
    a) If the number of numbers is odd sum of the numbers is odd 
    Ex:- 3 + 5 + 9 = 17(odd)
    b) If the number of numbers is even the sum is even
    Ex:- 3 + 5 + 7 + 9 =24(even)
4.    If the product of a certain number of numbers is odd, then none of the numbers is even 
     (odd)
    i.e the product of any number of odd numbers is odd.
5.    If the product of certain number of numbers is even, then atleast one of them is even.
     7 x 5 x 10 = 750 
6.    If  is even number, then odd number is i.e.,  the difference of even and odd number is ‘1’
    i.e. 2n- (2n -1) = 2n - 2n +1 =1