Prime Numbers

A prime number is a number that has only two factors, 1 and the number itself. For example, 2, 3, 5, and 7 are prime numbers because they cannot be divided evenly by any other number.

A number with more than two factors is called a composite number. For example, 4 is composite because it can be divided by 1, 2, and 4.
 

Here is the list of prime numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
 

Let us look at a few related types of prime numbers, and they are as follows:
 

Co-prime numbers:
Two numbers are called co-prime numbers if they have no common factors other than 1.
For example, (8, 15) are co-prime because their only common factor is 1.
 

Twin prime numbers:
Two prime numbers that have a difference of 2 are called twin primes.
For example, (3, 5), (5, 7), (11, 13), and (17, 19) are twin prime pairs.
 

Facts About Prime Numbers
Here are some essential facts about prime numbers that will help you understand and solve many mathematical problems easily:
 

The smallest prime number is 2, as it has only two factors, 1 and itself.
 

The largest known prime number (as of November 2022) is 2⁸²,⁵⁸⁹,⁹³³ − 1, which has 24,862,048 digits and was discovered by the Great Internet Mersenne Prime Search (GIMPS) in 2018.
 

2 is the only even prime number, while all other prime numbers are odd, so they are called odd prime numbers.
 

Twin prime numbers are pairs of prime numbers that differ by 2 or have one composite number between them. For example, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73).
 

Co-prime numbers are two numbers whose highest common factor (HCF) is 1, meaning they have no common factors other than 1. For example: (6, 13) are co-prime even though both are not prime.

Numbers can sometimes be either classified as prime or composite based on their factors.

Here are some important properties of prime numbers explained in simple terms
 

• Every number greater than 1 can be divided by at least one prime number.
   
• Every even positive number greater than 2 can be written as the sum of two prime numbers.
   
• Except for 2, all prime numbers are odd, which means 2 is the only even prime number.
   
• Any two prime numbers are always co-prime, as they have no common factor other than 1.
   
• Every composite number can be broken down into prime factors, and this factorization is unique for each number.

There are two simple ways to check if a number is prime or not.


Method 1:
Apart from 2 and 3, every prime number can be written in the form of 6n + 1 or 6n - 1, where n is a natural number. This rule helps identify many prime numbers, except those that are multiples of smaller primes such as 2, 3, 5, 7, or 11. For example, when n = 1, we get 5 and 7; when n = 2, we get 11 and 13; when n = 3, we get 17 and 19; when n = 4, we get 23 and 35, where 25 is not prime because it’s a multiple of 5, and when n = 5, we get 29 and 31, both of which are prime numbers.


Method 2:
Another formula to find prime numbers greater than 40 is n² + n + 41, where n can be any natural number from 0 to 39. For example, when n = 0, the result is 41; when n = 1, the result is 43; when n = 2, the result is 47; when n = 3, the result is 53; when n = 4, the result is 61; and when n = 5, the result is 71. All these are prime numbers.

Confused about how to learn and master prime numbers? When learning prime numbers, try to follow these tips and tricks to master them. These tricks can help you learn prime numbers quickly. 

• All even numbers except 2 are non-prime because they have more than 2 factors. For example, 256 is not a prime number, as it has more than two factors. The common factor between any two prime numbers is 1. So all prime numbers are co-primes.
  
   
• The two prime numbers with a difference of 2 are twin primes. 
  
   
• All composite numbers can be expressed as a product of prime numbers.
  
   
• No prime numbers end with 5 except five, as it is divisible by 5.
  
   
• The numbers ending with 0 are not prime numbers, as they are divisible by 2 and 10.
  
   
• All the even numbers except two can be expressed as the sum of two odd numbers.
  
   
• As a parent, you can encourage your child to explore prime and non-prime numbers through everyday examples, such as counting the fruits in a basket, the fingers on a hand, or the coins in a jar.
   
• Relate the concept of primes to real-world applications, such as cryptography, patterns, or coding activities.

In our daily lives, we use prime numbers in various fields, such as cryptography and data security, as well as in analyzing number patterns and other applications. Let’s learn a few real-world applications of prime numbers.

 
Cryptography and data security: Prime numbers are used in encryption algorithms, such as RSA, which uses public-key cryptography. Cryptography is used to decrypt or encrypt data. For data security, prime numbers can be used as passwords.


Simplification of fractions: Prime numbers can be used to find the GCF of more than two numbers. GCF is used to simplify the fractions.


Analyzing number patterns: As prime numbers are considered the building blocks of all integers, they help understand the relationship between numbers. We use prime numbers in hash tables to evenly distribute data keys. 


Computer programs and games: Prime numbers are used in computer programs to make random numbers. These random numbers help make games, apps, and secret codes safer.


Music and sounds: Prime numbers are used in music and sound systems to make sounds clearer and smoother. They also help in creating music patterns that don’t repeat often.