Is 735 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 735 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 735 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers, such as: 

• 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 number of 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 735 is prime or composite.

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

Step 2: Divide 735 by 3. It is divisible by 3, so 3 is a factor of 735.

Step 3: Divide 735 by 5. It is not divisible by 5, so 5 is not a factor of 735.

Step 4: You can simplify checking divisors up to 735 by finding the square root. We then need to only check divisors up to the square root value.

Step 5: When we divide 735 by 3, 5, 7, and 11, it is divisible by 3, 5, and 7.

Since 735 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 5, which is not even. Therefore, 735 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 735 is 15. Since 15 is divisible by 3, 735 is also divisible by 3. 

Divisibility by 5: The unit’s place digit is 5. Therefore, 735 is divisible by 5. 

Divisibility by 7: The last digit in 735 is 5. To check divisibility by 7, double the last digit (5 × 2 = 10). Then, subtract it from the rest of the number (73 - 10 = 63). Since 63 is divisible by 7, 735 is also divisible by 7. 

Divisibility by 11: In 735, the sum of the digits in odd positions is 10, and the sum of the digits in even positions is 3. The difference is 7, which is not divisible by 11.

Therefore, 735 is not divisible by 11. Since 735 is divisible by 3, 5, and 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 these steps:

Step 1: Write numbers from 1 to 1000 in rows and columns.

Step 2: Leave 1 unmarked, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all multiples of 3.

Step 5: Repeat this process until you have marked and crossed numbers up to 1000.

Through this process, we will have a list of prime numbers from 1 to 1000. The list does not include 735, 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 735 as 3 × 245.

Step 2: In 3 × 245, 245 is a composite number. Further, break down 245 into 5 × 49.

Step 3: In 5 × 49, 49 is a composite number. Break it down into 7 × 7.

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

Hence, the prime factorization of 735 is 3 × 5 × 7 × 7.

• 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.

• Prime numbers: Natural numbers greater than 1 that are only divisible by 1 and themselves. For example, 17 is a prime number.

• 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 the actual division.

• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.