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 1345 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 1345 has more than two factors, it is not a prime number. Several 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 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 1345 is prime or composite.

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

Step 2: Divide 1345 by 2. It is not divisible by 2 because it's an odd number.

Step 3: Divide 1345 by 3. The sum of the digits (1 + 3 + 4 + 5 = 13) is not divisible by 3, so 1345 is not divisible by 3.

Step 4: Divide 1345 by 5. The last digit is 5, so it is divisible by 5.

Hence, 5 is a factor of 1345. Since 1345 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: 1345 is an odd number, meaning it is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 1345 is 13. Since 13 is not divisible by 3, 1345 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 5. Therefore, 1345 is divisible by 5.

Divisibility by 7: No simple rule for 7 applies directly, but dividing 1345 by 7 gives a remainder, so it is not divisible by 7.

Divisibility by 11: The difference between the sum of the digits in odd positions (1 + 4 = 5) and even positions (3 + 5 = 8) is 3, which is not divisible by 11.

Thus, 1345 is not divisible by 11. Since 1345 is divisible by 5, 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 the following steps.

Step 1: Write numbers in a range, for example, from 1 to 1000, 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 until you reach the desired range consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers.

1345 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. Then multiply those factors to obtain the original number.

Step 1: We can write 1345 as 5 × 269.

Step 2: In 5 × 269, 269 is a prime number.

Step 3: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 1345 is 5 × 269.

• 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 numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves.

• Divisibility: The ability of one number to be divided by another without leaving a remainder.

• Factorization: The process of breaking down a number into its prime components.

• Co-prime numbers: Two numbers that have no common factors other than 1.