Is 409 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, such as:

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 409 has no divisors other than 1 and itself, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 409 has exactly two factors, it is a prime number. Several methods are used to distinguish between prime and composite numbers. Some of these 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 409 is prime or composite.

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

Step 2: Divide 409 by 2. It is not divisible by 2, so 2 is not a factor of 409.

Step 3: Continue checking divisibility by prime numbers up to the square root of 409.

Since 409 is not divisible by any number other than 1 and itself, 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: The number in the one's place value is 9, which is odd, so 409 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 409 is 13. Since 13 is not divisible by 3, 409 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 9, so 409 is not divisible by 5.

Divisibility by 7, 11, 13, etc.: Check divisibility by other prime numbers up to the square root of 409

Since 409 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 manageable range in rows and 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 with successive primes until you reach the desired range. Through this process, we will have a list of prime numbers.

Since 409 is not divisible by any smaller prime numbers, it remains marked as 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 409 by the smallest prime numbers, starting with 2, 3, 5, etc.

Step 2: Since 409 is not divisible by any prime numbers other than itself, it cannot be factored further.

Hence, 409 is a prime number because it cannot be broken down into smaller prime factors.

• Prime Numbers: Natural numbers greater than 1 that have only two factors, 1 and itself.
  
   
• Composite Numbers: Natural numbers greater than 1 that have more than two factors.
  
   
• Divisibility: A number's ability to be divided by another number without leaving a remainder.
  
   
• Prime Factorization: The process of expressing a number as a product of its prime factors.
  
   
• Sieve of Eratosthenes: A method to find all prime numbers up to a certain number by iteratively marking the multiples of each prime.