PRIME AND COMPOSITE NUMBERS

 PRIME AND COMPOSITE NUMBERS
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