Prime Numbers and Composite Numbers

Prime Numbers and Composite Numbers:

Prime Numbers:

A prime number is any positive integer larger than 1 with exactly two factors: 1 and itself.
OR
A prime number has NO factors other than 1 and itself.

Prime numbers are the building blocks of integers.

Example:

5 ÷ 1 = 5, 5 ÷ 5 = 1, 5 is not divisible by 2, 3, and 4.  So 5 is prime number.
7 is a prime because the only factors of 7 are 1 and 7.
However, 6 is not a prime because it is divisible by 1, 2, 3, 6.

1 is not considered as prime, as it has only one factor (itself).

Therefore, the first prime number is 2, which is also the only even prime.
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Composite Numbers:

A composite number is any positive integer larger than 1 which has at least one factor other than 1 and the number itself.
OR
A number other than 1 is called a composite number, if it is not prime.

1 is neither prime nor composite. It is the only number with this property.

Factors of few numbers:

Number	Factors	Number of factors
1	1	1
2	1, 2	2
3	1, 3	2
4	1, 2, 4	3
5	1, 5	2
6	1, 2, 3, 6	4
7	1, 7	2
8	1, 2, 4, 8	4
9	1, 3, 9	3
10	1, 2, 5, 10	4
11	1, 11	2
12	1, 2, 3, 4, 6, 12	6
13	1, 13	2
14	1, 2, 7, 14	4
15	1, 3, 5, 15	4

We can observe that numbers can be classified into following categories:

1) Numbers having exactly one factor.
2) Numbers having exactly two factors.
3) Numbers having more than two factors.

From the above results we can say that,
The natural number 1 is the only number which has exactly one factor, the number itself.
Each of the numbers 2, 3, 5, 7, 11, 13, etc., has exactly two factors, namely 1 and the number itself. Such numbers are called prime numbers.

Important Facts:

1) 1 is neither prime nor composite.
2) 2 is the lowest prime number.
3) 2 is the only even prime number. All other even numbers are composite numbers.
4) 4 is the smallest composite number.
5) 2, 3, 5, and 7 are prime numbers of one-digit.
6) 4, 6, 8 and 9 are the composite numbers of one-digit.

Finding Prime and Composite numbers from 1 to 100:

To find prime and composite numbers from 1 to 100 we follow the following steps:
Step 1) Prepare a table of numbers from 1 to 100.
Step 2) We know that 1 is neither prime nor composite. So, we separate it out by making a box around it.
Step 3) Encircle 2 as a prime number and cross out every multiple of 2.
Step 4) Encircle 3 as a prime number and cross out every multiple of 3. We need not mark the numbers which have already been crossed out.
Step 5) Encircle 5 as a prime number and cross out every multiple of 5. We need not mark the numbers which have already been crossed out.
Step 6) Continue this process till the numbers up to 100 are either encircled or crossed-out.
Step 7) List all encircled numbers in the table as prime numbers and all crossed out numbers as composite numbers.

In this case, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 are prime numbers between 1 and 100. The remaining numbers other than 1 are composite numbers.

From the above list of prime numbers between 1 and 100, we observe that every prime number other than 2 is odd but every odd number need not be prime.

Twin Primes:

Two prime numbers are known as twin—primes if there is only one composite number between them.
Pairs of twin-primes between 1 and 100 are:
3,5;   5,7;   11,13;   17,19;   29,31;   41,43;   59,61   and   71,73.

Prime Triplet:

A set-of three consecutive prime numbers, differing by 2, is called a prime triplet.
The only prime triplet is (3, 5, 7).

Co-Primes:

Two numbers are said to be co-prime or relatively prime if they do not have a common factor other than 1.
2,3;   3,4;   5,6;   8,13;   12,23, etc. are pairs of co-primes.

Any two prime numbers are always co—primes, but two co-primes need not be both prime numbers. For example, 14, 15 are co-primes, while none of 14 and 15 is a prime n umber.

Rule to check whether a number between 100 and 200 is prime or not:

Examine whether the given number is divisible by any prime number less than 15, i.e., 2, 3, 5, 7, 11 and 13. If it is divisible, then it is not prime; otherwise it is prime.

Example: Check whether the following numbers are prime numbers:
1) 179      2) 117      3) 139
Solution: 1) Given number is 179
We find that 179 is not completely divisible by any of the numbers 2, 3, 5, 7, 11 and 13. So, it is a prime number.
2) Given number is 117.
We find that 117 is divisible by 13. So, it is not a prime number.
3) Given number is 139
We find that 139 is not completely divisible by any of the numbers 2, 3, 5, 7, 11 and 13. So, it is a prime number.

Rule to check whether a number between 100 and 400 is prime or not:

If a number between 100 and 400 is not divisible by any prime number less than 20, i.e., 2, 3, 5, 7, 11, 13, 17 and 19, then it is prime; otherwise it is not prime.

Example: Check whether the following is a prime number.
1) 291     2) 323     3) 277
Solution: 1) Given number is 291.
In order to check whether it is prime or not, we check whether it is divisible by any one of the primes 2, 3, 5, 7, 11, 13, 17 and 19.
Clearly, 291 is completely divisible 3.
So, it is not a prime number.
2) We find that 323 is divisible by 17. So, it is not a prime number.
3) Clearly, 277 is not divisible by any of the prime numbers 2, 3, 5, 7, 11, 13, 17 and 19. So, it is a prime number.

Some Facts about Prime Numbers:

1) There are infinitely many primes.
We know that 2 is the first prime and 2 + 1 = 3 is also a prime. Now, 2 and 3 are primes and 2 x 3 + 1 = 7 is also a prime. Similarly, 2, 3 and 5 are primes and 2 x 3 x 5 + 1 = 31 is also a prime.
Similarly, we can generate more primes as given below:
2 x 3 x 5 x 7 + 1 = 211
2 x 3 x 5 x 7 x 11 + 1 = 2311
2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031, etc.
Thus, primes generate more primes. Hence, there are infinitely many primes.
2) Every prime number except 2 is an odd number.
3) A natural number greater than 1 is either a prime or it can be expressed as a product of primes.
4) There is no largest prime number.
5) If p and q are prime factors of a number a, then their product p x q is also a factor of a.
We observe that 2 and 5 are prime factors of 20 and 2 x 5 = 10 is also a factor of 20.
Similarly, 15 is a factor of 90 because 15 = 3 x 5 and and 5 are prime factors of 90.