Is 421 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 that is 1.

As 421 has only two factors, it is indeed a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 421 has exactly two factors, it is a prime number. Various methods can be used to determine whether a number is prime or composite. 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 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 421 is prime or composite.

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

Step 2: Check divisibility by numbers less than the square root of 421, which is approximately 20.5.

Step 3: 421 is not divisible by any number other than 1 and 421 itself.

Since 421 has exactly 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: 421 is not divisible because it is odd.

- Divisibility by 3: The sum of the digits in 421 is 7, which is not divisible by 3.

- Divisibility by 5: 421 does not end in 0 or 5, so it's not divisible by 5.

- Divisibility by 7: When checked, 421 is not divisible by 7.

- Divisibility by 11: The alternating sum of the digits is 3, which is not divisible by 11.

Since 421 is not divisible by any of these numbers, it confirms that 421 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 1 to 1000 in a grid format.

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 with all numbers up to the square root of the maximum number in the grid.

Through this process, we can see that 421 is not crossed out, affirming that it is a prime number.

Prime factorization is a process of breaking down a number into prime factors, then multiplying those factors to obtain the original number.

Step 1: Attempt to divide 421 by the smallest prime numbers (2, 3, 5, 7, 11, etc.).

Step 2: As 421 is not divisible by any prime number other than itself, it cannot be factored further.

Thus, 421 remains as 1 and 421, confirming it as a prime number.

• Prime numbers: Natural numbers greater than 1 having exactly two distinct positive divisors: 1 and themselves.
  
   
• Composite numbers: Natural numbers greater than 1 that have more than two distinct positive divisors.
  
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.
  
   
• Divisibility rules: Guidelines that help determine whether a number can be divided by another without performing division.
  
   
• Prime factorization: The expression of a composite number as a product of prime numbers.