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 752 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 752 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers. These 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 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 752 is prime or composite.

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

Step 2: Divide 752 by 2. It is divisible by 2, so 2 is a factor of 752.

Step 3: Divide 752 by 3. It is not divisible by 3, so 3 is not a factor of 752.

Step 4: You can simplify checking divisors up to 28 (approximately √752).

Step 5: When we divide 752 by 2, 4, 8, etc., it is divisible by these numbers.

Since 752 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 in the ones' place is 2, which is an even number. Therefore, 752 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 752 is 14. Since 14 is not divisible by 3, 752 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 2. Therefore, 752 is not divisible by 5.

Divisibility by 7: Double the last digit (2 × 2 = 4) and subtract it from the rest of the number (75 - 4 = 71). Since 71 is not divisible by 7, 752 is also not divisible by 7.

Divisibility by 11: In 752, the difference between the sum of the digits in odd positions (7 + 2 = 9) and the sum of the digits in even positions (5) is 4, which is not divisible by 11.

Since 752 is divisible by other numbers like 2 and 4, 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 these steps.

Step 1: Write 1 to 1000 in rows and columns.

Step 2: Leave 1 without coloring or crossing it, 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 have marked and crossed out all numbers except 1. Through this process, we will have a list of prime numbers.

752 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 752 as 2 × 376.

Step 2: In 2 × 376, 376 is a composite number. Further, break 376 into 2 × 188.

Step 3: Continue breaking down until all factors are prime: 2 × 2 × 2 × 94; then 94 becomes 2 × 47.

Step 4: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 752 is 2 × 2 × 2 × 2 × 47.

• 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 breaking down a number into its prime factors.

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

• Co-prime numbers: Two numbers that have only 1 as their common factor. For example, 8 and 15 are co-prime.

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