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.
   
• Since 743 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 743 has only two factors, it is indeed a prime number. Several methods can be 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 743 is prime or composite.

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

Step 2: Divide 743 by 2. It is not divisible by 2, so 2 is not a factor of 743.

Step 3: Divide 743 by 3. It is not divisible by 3, so 3 is not a factor of 743.

Step 4: You can simplify checking divisors up to 743 by finding the approximate square root value. We then need to only check divisors up to the square root value.

Step 5: When we divide 743 by numbers up to its square root, no other numbers divide it completely.

Since 743 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: The number in the ones' place value is 3. Since 3 is not even, 743 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 743 is 14. Since 14 is not divisible by 3, 743 is also not divisible by 3.

Divisibility by 5: The unit's place digit is 3. Therefore, 743 is not divisible by 5.

Divisibility by 7, 11, 13, etc., reveal no divisibility for these numbers.

Since 743 is not divisible by any numbers other than 1 and 743 itself, it 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 following steps.

Step 1: Write numbers in a sequence up to a certain limit.

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 limit.

Through this process, we will have a list of prime numbers in that range.Checking for 743 in such a list confirms it as a prime number since it is not crossed out.

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

Step 1: Attempt to divide 743 by the smallest prime numbers like 2, 3, 5, 7, etc.

Step 2: None of these divisions result in a whole number, confirming there are no prime factors other than 1 and 743 itself.

Thus, 743 is a prime number.

• Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.

• Composite Number: A natural number greater than 1 that has more than two positive divisors.

• Divisor: A number that divides another number completely without leaving a remainder.

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

• Co-prime Numbers: Two numbers that have only 1 as their common factor.