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 us 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 a prime number. For example, 7 has only 2 factors, 1 and 7 itself. So, it is a prime number. However, 6 has four factors, 1, 2, 3 and 6. Therefore, it is not a prime number. It is a composite number.
Prime Number Definition
Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number.
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:
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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 and 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. The following points list the difference between prime and coprime numbers.
 Coprime numbers are always considered in pairs, while a prime number is a single number.
 If a pair of numbers has no common factor apart from 1, then the numbers are called coprime numbers. Prime numbers do not have any such condition.
 Coprime numbers can be prime or composite, but the only condition is that the Greatest Common Factor (GCF) of coprime numbers is always 1. Prime numbers are different from composite numbers because prime numbers are those numbers that have only 2 factors, 1 and the number itself.
Examples of Coprime Numbers
 5 and 9 are coprimes. The factors of 5 are 1, 5; and the factors of 9 are 1, 3, and 9. We can see that they have only 1 as their common factor, therefore, they are coprime numbers.
 6 and 11 are coprimes because they have only 1 as their common factor. The factors of 6 are 1, 2, 3 and 6; and the factors of 11 are 1 and 11. We can see that they have only 1 as their common factor, therefore, they are coprime numbers.
 18 and 35 are coprimes because they have only 1 as their common factor. The factors of 18 are 1, 2, 3, 6, 9 and 18; and the factors of 35 are 1, 5, 7 and 35. We can see that they have only 1 as their common factor, therefore, they are coprime numbers.
It should be noted that Coprime numbers need not necessarily be prime numbers.
Examples of Prime Numbers
A few examples of prime numbers are, 2, 3, 5, 7 and so on.
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 us substitute a few whole numbers and check.
 Let us check this formula, n^{2} + n + 41, for the number 0, So, 0^{2} + 0 + 41 = 0 + 41 = 41
 Let us check this formula, n^{2} + n + 41, for the number 1, So, 1^{2} + 1 + 41 = 2 + 41 = 43
 Let us check this formula, n^{2} + n + 41, for the number 2, So, 2^{2} + 2 + 41 = 6 + 41 = 47
Therefore, we can calculate all the prime numbers greater than 40 using this formula.
Method 2: Every prime number, apart from 2 and 3, can be written in the form of '6n + 1 or 6n  1'. So, if we have any number different from 2 and 3, we can check if it is prime or not by trying to express it in the form of 6n + 1 or 6n  1
 Let us check this formula, 6n  1, for the number 1, So, 6(1)  1 = 5
 Let us check this formula, 6n + 1 for the number 01, So, 6(1) + 1 = 7
 Let us check this formula, 6n  1, for the number 0, So, 6(2)  1 = 11
 Let us check this formula, 6n + 1, for the number 0, So, 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. It should be noted that all prime numbers are odd numbers except for the number 2 which is an even number. Interestingly, 2 is the only prime number which is even. This means the list of odd numbers can start from 3 onwards and continue because the rest of the prime numbers are odd numbers. For example, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and so on are odd prime numbers.
History of Prime Numbers
Prime numbers have 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? Eratosthenes was one of the greatest scientists who lived a few decades after Euclid. He designed a smart way to determine all the prime numbers up to a given number. This method is called the Sieve of Eratosthenes. Let us learn about the Sieve of Eratosthenes in the following section.
Sieve of Eratosthenes
The Sieve of Eratosthenes is an ancient algorithm that helps to find prime numbers up to any given limit. The following figure shows the prime numbers up to 100 that are found using the Sieve of Eratosthenes. The uncrossed numbers in the figure represent the prime numbers that are left after using the Sieve of Eratosthenes.
Suppose we need to find the prime numbers up to 'n', so we will generate the list of all numbers from 2 to n. The following steps show how to find all the prime numbers up to 100 using the Sieve of Eratosthenes.
 Step 1: First create a list of numbers from 2 to 100 as shown above. We leave the number 1 because all prime numbers are more than 1.
 Step 2: We start from the first number 2 in the list. We cross out every number which is a multiple of 2 except 2. For example, we cross 4, 6, 8, 10, 12, 14, 16, and so on up to 100.
 Step 3: Then, move on to the next unmarked (uncrossed) number which is 3. Cross out every number which is a multiple of 3 except 3. This will include a few multiples that are already crossed. So we cross the remaining ones. For example, 9, 15, 21, and so on up to 100.
 Step 4: Then, we move to the next uncrossed number in the list after 3 which is 5. Cross out every number which is a multiple of 5 except 5. This will include a few multiples that are already crossed. So we cross the remaining ones. For example, 25, 35, 55, and so on up to 100.
 Step 5: 7 is the next uncrossed number in the list after 5. So, cross out every number which is a multiple of 7 except 7. This will include a few multiples that are already crossed. So we cross the remaining ones. For example, 49. After this step, we will be left with only prime numbers. In other words, the Sieve of Eratosthenes filters out all numbers and only prime numbers are left behind. Observe the numbers that are crossed and that are left as prime numbers in the Sieve of Eratosthenes given above.
Smallest Prime Number
The smallest prime number is 2. Interestingly, this is the only even number that is a prime number. It should be noted that 1 is neither a prime number nor a composite number because it has only 1 factor which is 1, hence, it is a unique number.
Even Prime Number
The only prime number which is an even number is 2. This means all even numbers are composite numbers. 2 is a prime number because it has only 2 factors 1 and the number 2 itself.
Largest Prime Number
The largest prime number cannot be listed because numbers are infinite and the largest prime number cannot be determined.
Important Tips on Prime Numbers
 The only even prime number is 2.
 The first five prime numbers are 2, 3, 5, 7, and 11.
 All prime numbers are not odd. This is because 2 is an even number and a prime number as well. In fact, 2 is the only even prime number.
 1 is neither a prime number nor a composite number.
ā Topics Related to Prime Numbers
Prime Number Examples

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: State true or false with respect to prime numbers.
a.) 1 is a prime number.
b.) The only even prime number is 2.
c.) The first five prime numbers are 2, 3, 5, 7, and 9.
d.) All prime numbers are odd.
Solution:
a.) False, 1 is neither a prime number nor a composite number.
b.) True, the only even prime number is 2.
c.) False, the first five prime numbers are 2, 3, 5, 7, and 11.
d.) False, all prime numbers are not odd because 2 is a prime number but it is even. Interestingly, 2 is the only even prime number.
FAQs on Prime Numbers
What are Prime Numbers in Math?
Prime numbers are those numbers that have only two factors, i.e., 1 and the number itself. For example, 2, 3, 7, 11, and so on are prime numbers. On the other hand, numbers with more than 2 factors are called 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.
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 set of numbers whose Highest Common Factor (HCF) 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 cannot be negative.
What are Twin Prime Numbers?
Twin prime numbers are those pairs of prime numbers that have a difference of 2 between them. For example, 3 and 5 are twin prime numbers because 5  3 = 2. These pairs of numbers always have one composite number between them. In this case, 4 is a composite number that comes in between 3 and 5. Other examples of twin prime numbers are (5, 7), (11, 13) and so on.
Which of the Prime Numbers are Even Numbers?
2 is the only prime number which is an even number. This means all prime numbers are odd except 2 which is an even number.
Which Numbers are not Prime Numbers?
The numbers that are not prime numbers are called composite numbers. Composite numbers are those numbers that have more than 2 factors. For example, 4 is a composite number because it has three factors, 1, 2, and 4. Similarly, 44 is a composite number because it has six factors, 1, 2, 4, 11, 22 and 44.
Are Prime Numbers Odd?
No, all prime numbers are not odd. 2 is a prime number that is even, therefore, it cannot be said that all prime numbers are odd. Interestingly, 2 is the only prime number that is an even number.
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