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 1339 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 1339 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers: 

• 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 1339 is prime or composite.

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

Step 2: Divide 1339 by 2. It is not divisible by 2, so 2 is not a factor of 1339.

Step 3: Divide 1339 by 3. It is not divisible by 3, so 3 is not a factor of 1339.

Step 4: You can simplify checking divisors up to 1339 by finding the square root value. We then need to only check divisors up to the square root value. The square root of 1339 is approximately 36.57.

Step 5: Continue checking with divisors 5, 7, 11, etc., and find that 1339 is divisible by 7 and 191.

Since 1339 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 number in the ones' place value is 9. Since it is odd, 1339 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 1339 is 16. Since 16 is not divisible by 3, 1339 is not divisible by 3. 

Divisibility by 5: The unit’s place digit is 9. Therefore, 1339 is not divisible by 5.

Divisibility by 7: 1339 divided by 7 gives a whole number (191), so it is divisible by 7. 

Divisibility by 11: Alternate sum of digits gives 0, so 1339 is not divisible by 11.

Since 1339 is divisible by numbers other than 1 and itself, it has more than two factors. Therefore, it is a composite number.

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

Step 1: Write numbers in a list and systematically eliminate non-prime numbers. 

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite. 

Step 3: For each prime number, cross out all its multiples.

Through this process, we will have a list of prime numbers. 1339 does not appear in the list of prime numbers, confirming it is a composite 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: We can break down 1339 as 7 × 191. 

Step 2: Both 7 and 191 are prime numbers.

Hence, the prime factorization of 1339 is 7 × 191.

•  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 it is divisible by 1, 2, 3, 4, 6, and 12. 

• Prime numbers: Numbers greater than 1 that have no divisors other than 1 and themselves. For example, 5 is a prime number. 

• Divisibility: A number is divisible by another if it can be divided exactly without leaving a remainder. For example, 20 is divisible by 5. 

• Prime factorization: The process of expressing a number as a product of its prime factors. For example, 18 as 2 × 3 × 3. 

• Co-prime numbers: Two numbers that have no common factors other than 1. For example, 8 and 15 are co-prime.