Prime Numbers
Prime numbers are the numbers that have only two factors, that are, 1 and the number itself. Consider an example of number 5, which has only two factors 1 and 5. This means it is a prime number. Let's take another example of the number 6, which has more than two factors, i.e 1, 2, 3, and 6. This means 6 is not a prime number. Now, if we take the example of the number 1, we know that it has only one factor. So, it cannot be a prime number as a prime number should have exactly two factors. This means 1 is neither a prime nor a composite number, it is a unique number.
What Are Prime Numbers?
A number greater than 1 with exactly two factors, i.e. 1 and the number itself is defined as a prime number. In other words, if a number cannot be divided into equal groups, then it is a prime number. We can divide a number into groups with equal numbers of items/elements only if it can be factorized as a product of two numbers. For example, 7 cannot be divided into groups of equal numbers. This is because 7 can only be factorized as follows:
 7 × 1 = 7
 1 × 7 = 7
This means 1 and 7 are the only factors of 7. So, 7 is a prime number because it could not be divided into groups of equal numbers.
Definition of a Prime Number: Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number.
History of Prime Numbers
Prime numbers created human curiosity since ancient times. Even today, mathematicians are trying to find prime numbers with mystical properties. Euclid proposed the theorem on prime numbers  There are infinitely many prime numbers.
Do you know all the prime numbers from 1 to 100? Did you check if each number is divisible by the smaller numbers? Then, you definitely invested a lot of time and effort. Eratosthenes was one of the greatest scientists, who lived a few decades after Euclid, designed a smart way to determine all the prime numbers up to a given number. This method is called the Sieve of Eratosthenes. Suppose you have to find the prime numbers up to n, we will generate the list of all numbers from 2 to n. Starting from the smallest prime number p = 2, we will strike off all the multiples of 2, except 2 from the list. Similarly, assign the next value of p which is a prime number greater than 2.
List of Prime Numbers
There are 25 prime numbers from 1 to 100. The complete list of prime numbers from 1 to 100 is given below:
List of Numbers  Prime Numbers 
Between 1 and 10  2, 3, 5, 7 
Between 11 and 20  11, 13, 17, 19 
Between 21 and 30  23, 29 
Between 31 and 40  31, 37 
Between 41 and 50  41, 43, 47 
Between 51 and 100 
53, 59, 61, 67, 71, 73, 79, 83, 89, 97 
ā Check:
Check out few more interesting articles related to prime numbers for better understanding.
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 Prime Factorization
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 Prime Numbers Worksheets
Properties of Prime Numbers
Some of the important properties of prime numbers are given below:
 A prime number is a whole number greater than 1.
 It has exactly two factors, that is, 1 and the number itself.
 There is only one even prime number, that is, 2.
 Any two prime numbers are always coprime to each other.
 Every number can be expressed as the product of prime numbers.
Prime Numbers vs Composite Numbers
 A prime number is a number greater than 1 that has exactly two factors, while a composite number has more than two factors. For example, 5 can be factorized in only one way, that is, 1 × 5 (OR) 5 × 1. It has only two factors, which are, 1 and 5. Therefore, 5 is a prime number.
 A composite number is a number greater than 1 that has more than two factors. For example, 4 can be factorized in multiple ways. So, the factors of 4 are 1, 2, and 4. It has more than two factors. Therefore, 4 is a composite number. Let us understand the difference between prime numbers and composite numbers with the help of a table given below:
Prime Numbers  Composite Numbers 

Numbers, greater than 1, having only two factors, 1 and the number itself  Numbers greater than 1 having at least three factors 
2 is the smallest and the only even prime number  4 is the smallest composite number 
Examples of prime numbers are 2, 3, 5, 7, 11, 13, etc  Examples of composite numbers are 4, 6, 8, 9, 10, etc. 
Prime Numbers and Coprime Numbers
There is a difference between prime numbers and coprime numbers. Coprime numbers are always considered in pairs, while a single number can be interpreted as a prime number. If a pair of numbers has no common factor apart from 1, then the numbers are called coprime numbers. Coprime numbers can be prime or composite, the only criteria to be met is that the GCF of coprime numbers is always 1.
Examples of coprime numbers:
 5 and 9 are coprimes.
 6 and 11 are coprimes.
 18 and 35 are coprimes.
Coprime numbers need not necessarily be prime numbers.
Easy Way to Find Prime Numbers
There are different ways to find prime numbers. Let us go through two of these methods.
Method 1: Substitute whole numbers for n in the formula 'n^{2} + n + 41'. This formula will give you all the prime numbers greater than 40. Let's substitute a few whole numbers and check.
 0^{2} + 0 + 41 = 0 + 41 = 41
 1^{2} + 1 + 41 = 2 + 41 = 43
 2^{2} + 2 + 41 = 6 + 41 = 47
Continuing like this, you can calculate all the prime numbers greater than 40.
Method 2: Every prime number, apart from 2 and 3, can be written in the form of '6n + 1 or 6n  1'. So, if you have any number different from 2 and 3, you can check if it is prime or not by trying to express it in the form of 6n + 1 or 6n  1
 6(1)  1 = 5
 6(1) + 1 = 7
 6(2)  1 = 11
 6(2) + 1 = 13
Now, we know that the numbers 5, 7, 11, and 13 are prime.
List of Odd Prime Numbers
A prime number chart is a chart that shows the list of prime numbers in a systematic order. Given below is the prime number chart from numbers 1 to 100 that shows the list of odd prime numbers(highlighted in yellow)
Is There a Pattern in Prime Numbers?
The set of prime numbers between any two numbers can be found by following a pattern. The following figure shows a few prime numbers encircled and striking off all the numbers divisible by these prime numbers. You can follow this pattern until you reach the square root of the larger number, that is, 100
Topics Related to Prime Numbers:
Examples of Prime Numbers

Example 1: Which of the two numbers is a prime number, 13 or 15?
Solution
The number 15 has more than two factors: 1, 3, 5, and 15. Hence, it is a composite number. On the other hand, 13 has just two factors: 1 and 13. Hence, it is a prime number. Therefore, 13 is a prime number.

Example 2: Why is 20 not a prime number?
Solution:
The factors of 20 are 1, 2, 4, 5, 10, and 20. Thus, 20 has more than two factors. Since the number of factors of 20 is more than two numbers, it is NOT a prime number (20 is a composite number).

Example 3: Is 1331 a prime number or a composite number?
Solution:
To check whether 1331 is a prime number or a composite we need to identify its factors first. The factors of 1331 are 1, 11, 121, 1331. Thus, 1331 has more than two factors. Since the number of factors of 1331 is more than two numbers, it is NOT a prime number. Hence, 1331 is a composite number.
FAQs on Prime Numbers
What Are Prime Numbers in Math?
Numbers having only two factors i.e. 1 and the number itself are called prime numbers whereas numbers with more than 2 factors are composite numbers.
What Are the Examples of Prime Numbers?
A number greater than 1, with exactly two factors, 1 and itself, is defined as a prime number in mathematics. Here are some examples of prime numbers: 2, 3, 31, 101, 149, etc.
How to Find Prime Numbers?
One of the easiest methods to find that a given number p, is a prime number, is to check the number of factors of the number p. If p has exactly two factors, 1 and p, then we say that p is a prime number.
Why Is 2 a Prime Number?
The factors of 2 are 1 and 2. Since 2 has exactly two factors, therefore it is a prime number. Interestingly, 2 is the only even prime number and the smallest prime number.
What Are all the Prime Numbers from 1 to 100?
The prime numbers from 1 to 100 are 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
What is the Formula to Find Prime Numbers?
Suppose you have a number p and n is the smallest natural number such that n^{2} ≥ p. If p is divisible by any number less than or equal to n, then p is not prime otherwise, p is prime.
Which Is the Smallest Prime Number?
The smallest number which has exactly two factors 1 and itself is 2. Therefore, the smallest prime number is 2.
What Is the Difference Between a Prime and a Coprime Number?
A prime number is a number that has only two factors, that is, 1 and the number itself. For example, 2, 3, 5, 7 are prime numbers. Coprime numbers are the numbers whose highest common factor is 1. For example, 2 and 3 are coprime numbers.
Can Prime Numbers be Negative?
The prime numbers should be only whole numbers, and all the whole numbers are greater than 1. Therefore, a prime number can't be negative.
Which Is the Largest known Prime Number?
The largest prime number is 2^{82,589,933 } 1, which is having 24,862,048 digits.
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