Is 401 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 401 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 401 has exactly 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 numbers as prime or composite.

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 401 is prime or composite.

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

Step 2: Divide 401 by 2, 3, 5, 7, 11, 13, 17, 19. It is not divisible by any of these numbers.

Step 3: You can simplify checking divisors by finding the root value, which is approximately 20. We then need to only check divisors up to the root value.

Since 401 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: 401 is odd, so it is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 401 is 5. Since 5 is not divisible by 3, 401 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 1. Therefore, 401 is not divisible by 5.

Divisibility by 7, 11, 13, 17, 19: You can attempt division, and you will find that 401 is not divisible by any of these.

Since 401 is not divisible by any numbers 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 range, such as 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.

The list includes 401, so 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 401 by the smallest prime numbers like 2, 3, 5, 7, 11, 13, 17, and 19.

Step 2: As 401 is not divisible by any of these, 401 itself is a prime number.

Step 3: There are no prime factors other than 401 itself, confirming it is prime.

• Prime numbers: Numbers that have only two distinct positive divisors, 1 and the number itself.
   
• Composite numbers: Natural numbers greater than 1 that are divisible by more than two numbers.
   
• Factors: The numbers that divide the number exactly without leaving a remainder are called factors.
   
• Divisibility rules: A set of guidelines used to determine whether one number is divisible by another.
   
• Prime factorization: The process of determining which prime numbers multiply together to make the original number.