Is 723 a Prime Number?

There are two types of numbers, mostly

• Prime numbers
   
• 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 723 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 723 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers:

• 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 numbers as prime or composite.

• 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 723 is prime or composite.

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

Step 2: Divide 723 by 2. It is not divisible by 2, so 2 is not a factor of 723.

Step 3: Divide 723 by 3. It is divisible by 3, so 3 is a factor of 723.

Step 4: You can simplify checking divisors up to 723 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 723 by 3 and continue with other numbers (like 5, 7, etc.), we find more divisors. Since 723 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. This is called the Divisibility Test Method.

Divisibility by 2: 723 is odd, so it is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 723 is 12 (7+2+3), which is divisible by 3. So, 723 is divisible by 3.

Divisibility by 5: The unit’s place digit is 3, not 0 or 5. Therefore, 723 is not divisible by 5.

Divisibility by 7: Perform the divisibility test for 7 to find it is not divisible by 7.

Divisibility by 11: Perform the test and see that 723 is not divisible by 11. Since 723 is divisible by 3 and potentially other factors, it has more than two factors, making it 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 these steps:

Step 1: Write numbers in a sequence.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark primes up to a certain number (like 100) and cross out the multiples of these primes.

Step 4: Continue this process to list prime numbers. Through this process, we find that 723 is not on 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 multiplying those factors to obtain the original number.

Step 1: We can write 723 as a product of its factors.

Step 2: Find that 723 = 3 × 241, where both 3 and 241 are primes.

Step 3: The prime factors of 723 are 3 and 241.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 723 is a composite number because it is divisible by 1, 3, 241, and 723.
   
• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 3 is a prime number.
   
• Divisors: Numbers that divide a given number exactly without leaving a remainder. For example, the divisors of 6 are 1, 2, 3, and 6.
   
• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 723 is 3 × 241.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer, by systematically eliminating the multiples of each prime starting from 2.