Is 491 a Prime Number?

There are two types of numbers, mostly —

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 numbers because they have only one common factor, which is 1.

As 491 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 491 has only two factors, it is a prime number. Few methods are used to distinguish between prime and composite numbers. A few methods are:

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

The method in which we count the number of divisors to categorize the 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 would be prime. If the count is more than 2, then the number is composite. Let’s check whether 491 is prime or composite.

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

Step 2: When we check the numbers up to the square root of 491, which is approximately 22.14, we find that 491 is not divisible by any number except 1 and 491 itself.

Since 491 has only 2 divisors, it is a prime 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: 491 is odd, so it is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 491 is 14. Since 14 is not divisible by 3, 491 is not divisible by 3.

Divisibility by 5: The unit’s place digit is 1. Therefore, 491 is not divisible by 5.

Divisibility by 7: Applying the divisibility rule of 7, 491 is not found to be divisible by 7.

Divisibility by 11: In 491, the alternating sum of the digits does not result in a number divisible by 11, so 491 is not divisible by 11.

Since 491 is not divisible by any number other than 1 and itself, it is a prime 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 in a grid format, like from 1 to 500.

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.

Since 491 is not crossed out, it is a prime 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: Attempt to divide 491 by prime numbers like 2, 3, 5, 7, 11, 13, 17, and so on, up to the square root of 491.

Step 2: Since none of these prime numbers divide 491 exactly, 491 has no prime factors other than 1 and 491 itself. Hence, the prime factorization of 491 is 491.

Since it cannot be broken down into other prime numbers, 491 is a prime number.

• 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.
  
   
• Prime numbers: Numbers greater than 1 with no divisors other than 1 and itself, like 491.

• Divisibility rules: Guidelines to quickly determine if one number is divisible by another without performing full division.

• Prime factorization: Expressing a number as a product of prime numbers.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to any given limit.