Is 768 a Prime Number?

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 768 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 768 has more than two factors, it is not a prime number. Several 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 768 is prime or composite.

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

Step 2: Divide 768 by 2. It is divisible by 2, so 2 is a factor of 768.

Step 3: Divide 768 by 3. It is divisible by 3, so 3 is a factor of 768.

Step 4: You can simplify checking divisors up to 768 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 768 by multiple numbers such as 2, 3, 4, 6, 8, 12, 16, 24, etc., it is divisible by these numbers. Since 768 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 8. Since 8 is an even number, 768 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 768 is 21. Since 21 is divisible by 3, 768 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 8. Therefore, 768 is not divisible by 5.

Divisibility by 7: Apply the rule of 7 to check divisibility. 768 divided by 7 is 109.714, showing it is not divisible evenly.

Divisibility by 11: Calculate the alternating sum of the digits for divisibility by 11. 7 - 6 + 8 = 9, which is not divisible by 11.

Since 768 is divisible by multiple numbers such as 2 and 3, 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. 768 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 768 as 2 × 384.

Step 2: Further break 384 into composite parts: 384 as 2 × 192.

Step 3: Continue breaking down 192 into 96, 48, 24, 12, 6, and 3.

Step 4: The prime factorization process yields 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3.

Hence, the prime factorization of 768 is 2^8 × 3.

• 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 only two factors, 1 and the number itself. For example, 5 is a prime number as it is divisible only by 1 and 5.
   
• Factors: The numbers that divide the 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.
   
• Divisibility rules: Guidelines that help determine if one number is divisible by another without performing full division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7.