# Prime Numbers

Prime numbers are natural numbers greater than 1, having exactly two factors, 1 and the number itself. It can also be defined as a whole number that can’t be exactly divided by anything except 1 and itself. Thus negative numbers cannot be prime.

They are important in mathematics because they are the building blocks of the natural numbers. 2, 3, 5, 7, 11, 13, 17, 19 are a few examples of prime numbers.

Natural numbers greater than 1 that are not prime are known as composite numbers.

0 is not a prime number, as it has infinite factors. Similarly, 1 is also not a prime number as it has only one factor (the number 1 itself).

## List

A list of prime numbers from 1 to 100 is provided below for reference:

Thus there is a total of 25 prime numbers between 1 and 100. 2 is the only even prime number, and the rest are all odd. Again, 2 and 3 are the only two consecutive prime numbers.

## Chart

Eratosthenes’s method of finding prime numbers resulted in the formation of charts. Even today, they are helpful for our direct applications and mathematical calculations.

## How to Find Prime Numbers

There are different ways of finding prime numbers. They are:

### Method 1: The Sieve of Eratosthenes

Before the discovery of calculators and computers, this method was the most widely used for finding prime numbers. It was named after its discoverer Eratosthenes.

Here the steps show how to identify prime numbers between 1 and 100.

1. All the numbers from 1 to 100 are listed. 1 is left out as it is neither prime nor a composite number.
2. We start at the first prime number, 2, and encircle it. All the multiples of 2 except 2 are crossed out as those are not prime numbers. For example, 4, 6, 8, 10, 12, and so on, up to 100 are not prime.
3. The first uncrossed number next to 2, i.e., 3, is encircled. All the multiples of 3, except 3, are crossed out. As few multiples like 6, 12, 18, 24, 30 … are already crossed out, the remaining ones, like 9, 15, 21, 27, 33, and so on, up to 100, are crossed out now.

The process repeats until all the numbers on the list are either encircled or crossed out.

Thus, we obtain a list with all the encircled prime numbers, 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. We can also verify the list from the list of prime numbers above.

### Method 2

It is used to find all prime numbers except 2 and 3. Each prime number except the only consecutive prime numbers 2 and 3 can be represented as (6n + 1) or (6n – 1); here, n is a natural number.

For example, 11 can be written as (6 × 2) – 1, (here n = 1). So, 11 is a prime number.

Similarly,

31 = (6 ×5) + 1

37 = (6 ×6) + 1

53 = (6 ×9) – 1,

61 = (6 × 10) + 1  and so on.

Thus 11, 31, 37, 53, and 61 are prime numbers.

### Method 3

It is used to find prime numbers greater than 40. If a number greater than 40 can be expressed as (m2 + m +41), m = 0, 1, 2, … it is a prime number.

For example,

41 = 02 + 0 + 41

43 = 12 + 1 + 41

47 = 22 + 2 + 41

Thus, 41, 43, and 47 are prime numbers.

## Odd Numbers and Prime Numbers

Though all the prime numbers except 2 are odd, the converse is false.

For example, 5, 7, 11, and 13 are prime and odd numbers.

But if we consider 15, 21, and 27, all of them are odd numbers.

15 has 4 factors:  1, 3, 5 and 15

21 has 4 factors: 1, 3, 7, and 21

27 also has 4 factors 1, 3, 9 and 27

So these are not prime numbers.

## Solved Examples

Is 2023 a prime number?

Solution:

2023 can be written in the form (6n + 1) as (6 × 337) + 1
Thus it is a prime number.

Is 51 a prime number?

Solution:

Since 51 is divisible by 1, 3, 17, and 51, it has four distinct factors.
Also, 51 can be represented in the form of (6 × 8) + 3 or (6 × 9) – 3
Hence it is not of the form (6n ± 1)
So it is not a prime number.

E.g.3.

Which of the following numbers is a prime number?
49
12
53
39

Solution:

49 have 3 factors: 1, 7, and 49. Hence, it is not a prime number.
12 have factors 1, 2, 3, 4, 6, and 12. Thus it is not a prime number.
53 can be written as (6 × 9) – 1, which is of the form (6n – 1). Also, 53 has 2 factors 1 and 53. So, it is a prime number.
39 have factors 1, 3, 13, and 39. Thus it is also not a prime number. So, option (c) is correct.