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 744 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 744 has more than two factors, it is not a prime number. Few methods are 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 744 is prime or composite.

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

Step 2: Divide 744 by 2. It is divisible by 2, so 2 is a factor of 744.

Step 3: Divide 744 by 3. It is divisible by 3, so 3 is a factor of 744.

Step 4: You can simplify checking divisors up to 744 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 744 by 2, 3, 4, 6, 8, 12, etc., it is divisible by several numbers.

Since 744 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 4. Four is an even number, which means that 744 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 744 is 15. Since 15 is divisible by 3, 744 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 4. Therefore, 744 is not divisible by 5.

Divisibility by 7: Doubling the last digit (4 × 2 = 8) and subtracting it from the rest of the number (74 - 8 = 66), which is divisible by 7, means 744 is also divisible by 7.

Divisibility by 11: The sum of the digits in odd positions is 8, and the sum of the digits in even positions is 4. The difference is 4, which is not divisible by 11, so 744 is not divisible by 11.

Since 744 is divisible by 2, 3, 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 1000 in rows and 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. 744 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 744 as 2 × 372.

Step 2: In 2 × 372, 372 is a composite number. Further, break the 372 into 2 × 186.

Step 3: Continue breaking it down: 186 into 2 × 93, and 93 into 3 × 31.

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

Hence, the prime factorization of 744 is 2 × 2 × 2 × 3 × 31.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 744 is a composite number because 744 is divisible by 1, 2, 3, and several other numbers.

• Prime numbers: Numbers greater than 1 with only two factors, 1 and the number itself.

• Divisibility test: A set of rules to determine if one number is divisible by another without performing division.

• Prime factorization: Breaking down a number into the product of prime numbers.

• Sieve of Eratosthenes: An ancient algorithm to find all prime numbers up to a certain limit.