Prime Numbers and Co-Prime Numbers

Prime Numbers:-Number which does not have any factor other than 1 and number itself are called Prime Numbers.

Eg. 2, 3, 7, 11 ….

Large Prime Number can be calculated by the following method :

Method:
This method can be used to calculate primes larger then 100.

Procedure:
a) Calculate the whole number (w) greater then square root of n.
b) Find the prime numbers[p(x)] less than the whole number(w).
c) Now check if the given number n is divisible by any of the prime numbers p(x).
d) If it gets divided then the number n is not a prime number else it is a perfect prime number.

Example:
Validate if 131 is prime number.

Step1: Square root of 131 = 11.44
So rounding it to the nearest large whole number it becomes 12.

Step2: Prime Numbers less than 12 are 2,3,5,7,11

Step3:
Given number 131 is not divisible by 2 or 3 or 5 or 7 or 11
So the given number is a prime perfect number.

Composite Numbers:-
Numbers having more than 2 factors are known as composite numbers.

Eg. 4, 6, 8 …

Remember:-

1. 2 is only even prime number.
2. 4 is first composite number.
3. 1 is neither a prime nor a composite number.
4. Prime numbers are always in form of 6K+ 1 or 6K -1 but vice versa not true.

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 (p >3) the
p^2 = 6K + 1 or
p^2 = 24k + 1

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

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.

Co-Prime Numbers (Relatively Prime Numbers):-

Pair of numbers that do not have any common factor except 1 are known as Co-prime numbers.

Eg . (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.