Is 729 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 729 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 729 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. These 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 729 is prime or composite.

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

Step 2: Divide 729 by 3. It is divisible by 3, so 3 is a factor of 729.

Step 3: Divide 729 by 9. It is divisible by 9, so 9 is a factor of 729.

Step 4: You can simplify checking divisors up to 729 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 729 by 3, 9, and 27, it is divisible by these numbers. Since 729 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 9. Since 9 is not an even number, 729 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 729 is 18. Since 18 is divisible by 3, 729 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 9. Therefore, 729 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (9 × 2 = 18). Then, subtract it from the rest of the number (72 - 18 = 54). Since 54 is not divisible by 7, 729 is also not divisible by 7.

Divisibility by 11: In 729, the sum of the digits in odd positions is 16, and the sum of the digits in even positions is 2. The difference is not divisible by 11. Since 729 is divisible by numbers other than 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. The list includes 2, 3, 5, 7, 11, 13, and so on. 729 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 729 as 3 × 243.

Step 2: In 3 × 243, 243 is a composite number. Further, break 243 into 3 × 81.

Step 3: Now, break 81 into 3 × 27 and 27 into 3 × 9 and 9 into 3 × 3.

Step 4: Now we get the product consisting of only prime numbers. Hence, the prime factorization of 729 is 3 × 3 × 3 × 3 × 3 × 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.
   
• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is the square of 4.
   
• Divisibility: A number is divisible by another if the remainder is zero when divided.
   
• Prime factorization: A way of expressing a number as the product of its prime factors.
   
• Sieve of Eratosthenes: A method for identifying prime numbers up to a specified integer by eliminating multiples of each prime starting from 2.