Is 693 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 that is 1.

As 693 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 693 has more than two factors, it is not a prime number.

A 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 693 is prime or composite.

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

Step 2: Divide 693 by 2. It is not divisible by 2, so 2 is not a factor of 693.

Step 3: Divide 693 by 3. It is divisible by 3, so 3 is a factor of 693.

Step 4: Divide 693 by 7. It is divisible by 7, so 7 is a factor of 693.

Since 693 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 3, which is not even. Therefore, 693 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 693 is 18. Since 18 is divisible by 3, 693 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 3. Therefore, 693 is not divisible by 5.

Divisibility by 7: Divide 693 by 7 to check. It is divisible by 7, so 693 is divisible by 7. Since 693 is divisible by more than just 1 and itself, 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 from 1 to 1000.

693 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 693 as 3 × 231.

Step 2: In 3 × 231, 231 is a composite number. Further, break the 231 into 3 × 77.

Step 3: Break 77 into 7 × 11.

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

Hence, the prime factorization of 693 is 3 × 3 × 7 × 11.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 693 is a composite number because it is divisible by 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693.

• Twin primes: The pair of prime numbers whose difference is 2 are called twin prime numbers. For example, (11, 13) is a pair of twin primes because the difference of 13 and 11 is 2.

• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 8 are 1, 2, 4, and 8 because they divide 8 completely.

• Prime Numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 7 is a prime number because it can only be divided evenly by 1 and 7.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer. It systematically eliminates the multiples of each prime number starting from 2.