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 750 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 750 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 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 750 is prime or composite.

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

Step 2: Divide 750 by 2. It is divisible by 2, so 2 is a factor of 750.

Step 3: Divide 750 by 3. It is divisible by 3, so 3 is a factor of 750.

Step 4: You can simplify checking divisors up to 750 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 750 by 2, 3, 5, and 10, it is divisible by all.

Since 750 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 0. Zero is an even number, which means that 750 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 750 is 12. Since 12 is divisible by 3, 750 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 0. Therefore, 750 is divisible by 5.

Divisibility by 7: The last digit in 750 is 0. To check divisibility by 7, double the last digit (0 × 2 = 0). Then, subtract it from the rest of the number (75 - 0 = 75). Since 75 is divisible by 7, 750 is also divisible by 7.

Divisibility by 11: In 750, the sum of the digits in odd positions is 12, and the sum of the digits in even positions is 5. This would mean that 750 is not divisible by 11.

Since 750 is divisible by 2, 3, 5, and 7, 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 1 to 100 in 10 rows and 10 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 table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100.

The list is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. 750 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 750 as 2 × 375.

Step 2: In 2 × 375, 375 is a composite number. Further, break the 375 into 3 × 125.

Step 3: Now, break down 125 into 5 × 25.

Step 4: Finally, break down 25 into 5 × 5.

Step 5: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 750 is 2 × 3 × 5 × 5 × 5.

• 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 rules: Guidelines that help determine if one number is divisible by another without performing the division.

• Factors: The numbers that divide a number exactly without leaving a remainder are called factors. For example, the factors of 4 are 1, 2, and 4 because they divide 4 completely.

• Prime factorization: The process of expressing a number as the product of its prime factors.