Is 721 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 721 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 721 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. Some methods include:

• 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 721 is prime or composite.

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

Step 2: Divide 721 by 2. It is not divisible by 2, so 2 is not a factor of 721.

Step 3: Divide 721 by 3. The sum of the digits is 10, which is not divisible by 3, so 3 is not a factor of 721.

Step 4: Divide 721 by 7. It is divisible by 7, so 7 is a factor of 721.

Step 5: You can simplify checking divisors up to 721 by finding the root value, approximately 26.8, and only check divisors up to this value. Since 721 is divisible by 7 and more than two divisors exist, 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 1, which is not even. Hence, 721 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 721 is 10. Since 10 is not divisible by 3, 721 is not divisible by 3.

Divisibility by 5: The unit’s place digit is 1. Therefore, 721 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (1 × 2 = 2). Then, subtract it from the rest of the number (72 - 2 = 70). Since 70 is divisible by 7, 721 is divisible by 7.

Divisibility by 11: In 721, the sum of the digits in odd positions is 8, and the sum of the digits in even positions is 2. This would mean that 721 is not divisible by 11. Since 721 is divisible by 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 number 721. Through this process, we will have a list of prime numbers. 721 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 721 as 7 × 103.

Step 2: Both 7 and 103 are prime numbers.

Step 3: Thus, the prime factorization of 721 is 7 × 103.

• 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 it is divisible by 1, 2, 3, 4, 6, and 12.
   
• Divisibility rules: Guidelines to determine if a number is divisible by another without performing full division.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Sieve of Eratosthenes: An ancient algorithm for finding all prime numbers up to any given limit.
   
• 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.