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 751 has only two factors, it is indeed a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 751 has exactly two factors, it is a prime number. A few 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 numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize numbers: - 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 751 is prime or composite.

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

Step 2: Check divisors for 751 up to its square root (approximately 27.4).

Step 3: 751 is not divisible by any number other than 1 and itself.

Since 751 has only 2 divisors, it is a prime 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: 751 is odd, so not divisible by 2. 

Divisibility by 3: The sum of the digits of 751 is 13, which is not divisible by 3. 

Divisibility by 5: 751 does not end in 0 or 5, so it's not divisible by 5. 

Divisibility by 7: Doubling the last digit and subtracting from the rest gives 70. 70 is not divisible by 7. 

Continue similar checks for other small primes like 11, 13, 17, 19, 23. Since 751 is not divisible by any of these, it confirms 751 is a prime number.

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

Step 1: Write numbers in a systematic grid.

Step 2: Leave 1 without marking, as it is neither prime nor composite.

Step 3: Mark 2 and cross out all multiples of 2.

Step 4: Mark the next unmarked number (3) and cross out multiples.

Step 5: Repeat until the chart is completed up to a desired range. By using this chart, we can identify prime numbers.

While 751 isn't on a basic chart up to 100, calculating beyond shows that 751 is a prime number.

Prime factorization is the process of breaking down a number into its prime factors. Then multiply those factors to obtain the original number.

Step 1: Attempt to divide 751 by the smallest primes (2, 3, 5, 7, 11, etc.).

Step 2: None of these numbers divide 751 without a remainder.

The absence of divisibility by any primes confirms that 751 cannot be factored further, confirming it is a prime number.

•  Prime numbers: Natural numbers greater than 1 with no divisors other than 1 and themselves. 

• Composite numbers: Natural numbers greater than 1 that have more than two divisors. 

• Factors: Numbers that divide another number exactly without leaving a remainder. 

• Divisibility rules: Guidelines to determine if one number is divisible by another without performing full division. 

• Sieve of Eratosthenes: An ancient algorithm for finding all prime numbers up to any given limit.