There are two main types of numbers: prime numbers and composite numbers, which are categorized based on the number of factors they have.

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 have specific 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.
   
• Since 597 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 597 has more than two factors, it is not a prime number. There are several methods to distinguish between prime and composite numbers, including:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization

The method of counting the number of divisors to categorize numbers as prime or composite is called the counting divisors method. Based on the number of divisors, we categorize numbers as either prime or composite. If there is a total count of only 2 divisors, then the number is prime. If the count is more than 2, then the number is composite. Let’s check whether 597 is prime or composite.

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

Step 2: Divide 597 by 2. It is not divisible by 2, so 2 is not a factor of 597.

Step 3: Divide 597 by 3. It is divisible by 3, so 3 is a factor of 597.

Step 4: You can simplify checking divisors up to 597 by finding the square root value and only checking divisors up to this value.

Step 5: When dividing 597 by 3, 9, and other small divisors, it is divisible by 3.

Since 597 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. This is called the Divisibility Test Method.

Divisibility by 2: The number has an odd digit in the ones' place (7), so 597 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 597 is 21 (5 + 9 + 7 = 21). Since 21 is divisible by 3, 597 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 7, so 597 is not divisible by 5.

Divisibility by 7: Using the divisibility rule for 7, 597 is not divisible by 7.

Divisibility by 11: The alternating sum is 3 (5 - 9 + 7 = 3), so 597 is not divisible by 11.

Since 597 is divisible by 3, it has more than two factors, making it a composite number.

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

Step 1: Write numbers from 1 to 100 in a 10x10 grid.

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 multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all multiples of 3.

Step 5: Repeat this process until reaching the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100.

Since 597 is not on the list of prime numbers, 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 597 as 3 × 199.

Step 2: Both 3 and 199 are prime numbers.

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

Hence, the prime factorization of 597 is 3 × 199.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers. For example, 597 is a composite number because it is divisible by 1, 3, 199, and 597.

• Prime numbers: Numbers greater than 1 that have no divisors other than 1 and themselves. For example, 3 is a prime number.

• Factors: Numbers that divide a given number exactly without leaving a remainder. For example, the factors of 597 are 1, 3, 199, and 597.

• Divisibility rules: Guidelines to determine whether one number is divisible by another without performing the actual division.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 597 is 3 × 199.