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 1409 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 1409 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 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 1409 is prime or composite.

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

Step 2: Check divisibility by numbers starting from 2.

Step 3: Simplify checking divisors up to the square root of 1409.

Step 4: Upon checking, 1409 is divisible by 11.

Since 1409 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 last digit of 1409 is 9, which is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in 1409 is 14. Since 14 is not divisible by 3, 1409 is also not divisible by 3. 

Divisibility by 5: The unit’s place digit is 9, so 1409 is not divisible by 5. 

Divisibility by 7: There is no obvious simplification for 1409 with 7. 

Divisibility by 11: The alternating sum of its digits (1 - 4 + 0 - 9 = -12) is divisible by 11. Since 1409 is divisible by 11, it has more than two factors.

Therefore, it is a composite number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” In this method, we follow the following steps.

Step 1: Write numbers in a sequence, like from 1 to 1000.

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 for all numbers up to the square root of 1000. Through this process, we will have a list of prime numbers from 1 to 1000.

1409 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 and then multiplying those factors to obtain the original number.

Step 1: Start with the smallest prime number that divides 1409.

Step 2: 1409 is divisible by 11, resulting in 128.

Step 3: 128 can be broken down further into 2⁷.

Hence, the prime factorization of 1409 is 11 × 2⁷.

• 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 factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7.

• Divisibility rules: A set of rules that help determine if one number is divisible by another without performing division.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and itself. For example, 7 is a prime number.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specific integer.