Is 231 a Prime Number?

There are two main types of numbers: prime numbers and composite numbers, depending on the number of factors.

A prime number is a natural number that is divisible only by 1 and itself.

For example, 3 is a prime number because it is divisible by 1 and itself.
 

A composite number is a positive number that is divisible by more than two numbers.

For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers follow a few properties like:
 

• Prime numbers are positive numbers always greater than 1.
   
• 2 is the only even prime number.
   
• They have only two factors: 1 and the number itself.
   
• Any two distinct prime numbers are co-prime because they have only one common factor, which is 1.
   

As 231 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself.

Since 231 has more than two factors, it is not a prime number.

A few methods are used to distinguish between prime and composite numbers.

Some methods include:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization
   

The method in which we count the number of divisors to categorize numbers as prime or composite is called the counting divisors method.

Based on the count of the divisors, we categorize prime and composite numbers.

• If there is a total count of only 2 divisors, then the number is prime.
• If the count is more than 2, then the number is composite.
   

Let’s check whether 231 is prime or composite.
 

Step 1: All numbers are divisible by 1 and itself.

Step 2: Divide 231 by 2. It is not divisible by 2, so 2 is not a factor of 231.
 

Step 3: Divide 231 by 3. It is divisible by 3, so 3 is a factor of 231.
 

Step 4: Divide 231 by 11. It is not divisible by 11, so 11 is not a factor of 231. Since 231 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method.
 

Divisibility by 2: The number in the ones' place value is 1, which is not even. Therefore, 231 is not divisible by 2.
 

Divisibility by 3: The sum of the digits in the number 231 is 6. Since 6 is divisible by 3, 231 is also divisible by 3.
 

Divisibility by 5: The unit’s place digit is 1. Therefore, 231 is not divisible by 5.
 

Divisibility by 7: To check divisibility by 7, double the last digit (1 × 2 = 2). Then, subtract it from the rest of the number (23 - 2 = 21). Since 21 is divisible by 7, 231 is also divisible by 7.
 

Divisibility by 11: In 231, the difference between the sum of the digits in odd positions (2 + 1 = 3) and the sum of the digits in even positions (3) is 0, which is divisible by 11. Since 231 is divisible by 3, 7, and 11, it has more than two factors. Therefore, it is a composite number.

The prime number chart is a tool created by using a method called “The Sieve of Eratosthenes.”

In this method, we follow the following steps.

Step 1: Write numbers from 1 to 100 in 10 rows and 10 columns.
 

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.
 

Step 3: Mark 2 because it is a prime number and cross out all the multiples of 2.
 

Step 4: Mark 3 because it is a prime number and cross out all the multiples of 3.
 

Step 5: Repeat this process until you reach the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100. The list is 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. 231 is not present in the list of prime numbers, so it is a composite number.

Prime factorization is a process of breaking down a number into prime factors.

Then multiply those factors to obtain the original number.

Step 1: We can write 231 as 3 × 77.
 

Step 2: In 3 × 77, 77 is a composite number. Further, break the 77 into 7 × 11.
 

Step 3: Now we get the product consisting of only prime numbers. Hence, the prime factorization of 231 is 3 × 7 × 11.
 

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 12 is a composite number because 12 is divisible by 1, 2, 3, 4, 6, and 12.
   
• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 4 are 1, 2, and 4 because they divide 4 completely.
   
• Divisibility rules: Techniques used to determine if one number is divisible by another without performing division.
   
• Prime factorization: Breaking down a composite number into a product of its prime factors. Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves.