Prime Numbers

Prime numbers are the positive integers having only two factors, 1 and the integer itself. For example, factors of 6 are 1,2,3 and 6, which are four factors in total. We can also say that the prime numbers are the numbers, which are only divisible by 1 or the number itself. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers. There is no defined formula to find if a number is prime or not (except to a certain range), apart from finding its factors. However, the numbers such as 7 have only two factors, i.e. 1 and 7. Hence, 7 is a prime number, but 6 is not; instead, it is a composite number. But always remember that 1 is neither prime nor composite. 

Also, we can say that except 1, remaining numbers are classified as prime and composite numbers. In this article, you will learn what a prime number is, how to find prime numbers using different techniques and the list of prime numbers along with the chart and examples.

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What is a Prime Number?

A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself. Any number which does not follow this is termed as composite numbers, which means that they can be factored into other positive integers.

First Ten Prime Numbers

The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Note: It should be noted that 1 is a non-prime number.

History of Prime Numbers

The prime number was discovered by Eratosthenes (275-194 B.C., Greece). He took the example of a sieve to filter out the prime numbers from a list of natural numbers and draining out the composite numbers.

Students can practise this method, by writing the positive integers from 1 to 100 and circling the prime numbers and putting a cross mark to composite numbers.

List of Prime Numbers 1 to 100

As we know, the prime numbers are the numbers which have only two factors which are 1 and the number itself. There are a number of primes in the number system. Let us provide here the list of prime numbers that are present between 1 and 100, along with their factors and prime factorisation.

Also, get the list of prime numbers from 1 to 1000 here.

Prime Numbers Chart

Before calculators and computers, numerical tables were used for recording all of the primes or prime factorizations up to a specified limit and are usually printed. The most beloved method for producing a list of prime numbers is called the sieve of Eratosthenes. This method results in a chart called Eratosthenes chart, as given below. The chart below shows the list of prime numbers up to 100, represented in coloured boxes.

Even Prime Number

As we know, the prime numbers are the numbers that have only two factors and the numbers that are evenly divisible by 2 are even numbers. Therefore, 2 is the only prime number that is even and the rest of the prime numbers are odd numbers, hence called odd prime numbers. 

Twin Prime Numbers

The prime numbers that have only one composite number between them are called twin prime numbers or twin primes. The other definition of twin prime numbers is the pair of prime numbers that differ by 2 only. For example, 3 and 5 are twin primes because 5 – 3 = 2.

The other examples of twin prime numbers are:

• (5, 7)        [7 – 5 = 2]
• (11, 13)    [13 – 11 = 2]
• (17, 19)    [19 – 17 = 2]
• (29, 31)    [31 – 29 = 2]
• (41, 43)    [43 – 41 = 2]
• (59, 61)    [61 – 59 = 2]
• (71, 73)    [73 – 71 = 2]

Click here to learn more about twin prime numbers.

Coprime Numbers

The pair of numbers that have only one factor in common between them, are called coprime numbers. Prime factors and coprime numbers are not the same. For example, 6 and 13 are coprime because the common factor between them is 1 only.

How to Find Prime Numbers?

The following two methods will help you to find whether the given number is a prime or not.
Method 1:
We know that 2 is the only even prime number. And only two consecutive natural numbers which are prime are 2 and 3. Apart from those, every prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a natural number.
For example:
6(1) – 1 = 5
6(1) + 1 = 7
6(2) – 1 = 11
6(2) + 1 = 13
6(3) – 1 = 17
6(3) + 1 = 19
6(4) – 1 = 23
6(4) + 1 = 25 (multiple of 5)
…
Method 2:
To know the prime numbers greater than 40, the below formula can be used.
n² + n + 41, where n = 0, 1, 2, ….., 39
For example:
(0)² + 0 + 0 = 41
(1)² + 1 + 41 = 43
(2)² + 2 + 41 = 47
…..

Prime Numbers 1 to 200

Here is the list of prime numbers from 1 to 200, which we can learn and also crosscheck if there exist any other factors for them.

Properties of Prime Numbers

Some of the properties of prime numbers are:

• Every number greater than 1 can be divided by at least one prime number.
• Every even positive integer greater than 2 can be expressed as the sum of two primes.
• Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
• Two prime numbers are always coprime to each other.

Each composite number can be factored into prime factors and individually all of these are unique in nature.

Prime Numbers and Composite Numbers

Prime Numbers 1 to 1000

There are a total of 168 prime numbers between 1 to 1000. They are:

Is 1 a Prime Number?

Conferring to the definition of the prime number, which states that a number should have exactly two factors for it to be considered a prime number. But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered a Prime number.

Examples: 2, 3, 5, 7, 11, etc

In all the positive integers given above, all are either divisible by 1 or itself, i.e. precisely two positive integers.

Also, chek:

Is 97 a prime number? 

Is 61 a prime number? 

Is 57 a prime number? 

Is 83 a prime number?

Smallest Prime Number

The smallest prime number defined by modern mathematicians is 2. To be prime, a number must be divisible only by 1 and the number itself which is fulfilled by the number 2.

Largest Prime Number

As of January 2020, the largest known prime number is 2^(82,589,933) – 1 a number which has 24,862,048 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS) in 2018.

Prime Number Examples

Example 1:

Is 10 a Prime Number?

Solution:

No, because it can be divided evenly by 2 or 5, 2×5=10, as well as by 1 and 10.

Alternatively,

Using method 1, let us write in the form 6n ± 1. 

10 = 6(1) + 4 = 6(2) – 2

This is not of the form 6n + 1 or 6n – 1.

Hence, 10 is not a prime number.

Example 2:

Is 19 a Prime Number?

Solution:

Let us write the given number in the form of 6n ± 1.
6(3) + 1 = 18 + 1 = 19
Therefore, 19 is a prime number.

Example 3:

Find if 53 is a prime number or not.

Solution:

The only factors of 53 are 1 and 53.

Or

Let us write the given number in the form of 6n ± 1.

6(9) – 1 = 54 – 1 = 53

So, 53 is a prime number.

Example 4:

Check if 64 is a prime number or not.

Solution:

The factors of 64 are 1, 2, 4, 8, 16, 32, 64.

Hence, it is a composite number and not a prime number.

Practice Problems

1. Identify the prime numbers from the following numbers:
   34, 27, 29, 41, 67, 83
2. Which of the following is not a prime number?
   2, 19, 91, 57
3. Write the prime numbers less than 50.

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