Is 702 a Prime Number?

There are two main types of numbers —

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 702 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 702 has more than two factors, it is not a prime number. Several 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 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 702 is prime or composite.

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

Step 2: Divide 702 by 2. It is divisible by 2, so 2 is a factor of 702.

Step 3: Divide 702 by 3. It is divisible by 3, so 3 is a factor of 702.

Step 4: You can simplify checking divisors up to 702 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 702 by 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, it is divisible by all these numbers.

Since 702 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: The number in the ones' place value is 2. Two is an even number, which means that 702 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 702 is 9. Since 9 is divisible by 3, 702 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 2. Therefore, 702 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (2 × 2 = 4) and then subtract it from the rest of the number (70 - 4 = 66). Since 66 is not divisible by 7, 702 is also not divisible by 7.

Divisibility by 11: In 702, the sum of the digits in odd positions is 9, and the sum of the digits in even positions is 0. This means that 702 is not divisible by 11.

Since 702 is divisible by 2 and 3, it has more than two factors. Therefore, it is a composite number.

The prime number chart is a tool created 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 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.

702 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 and then multiplying those factors to obtain the original number.

Step 1: We can write 702 as 2 × 351.

Step 2: In 2 × 351, 351 is a composite number. Further, break 351 into 3 × 117.

Step 3: Now, break down 117 into 3 × 39, and 39 into 3 × 13.

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

Hence, the prime factorization of 702 is 2 × 3 × 3 × 3 × 13.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 702 is a composite number because it is divisible by more than two numbers.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Divisibility Test: A method to determine if one number is divisible by another without performing the actual division.

• Factors: The numbers that divide a given number exactly without leaving a remainder are called factors.

• Co-prime numbers: Two numbers are co-prime if their greatest common divisor is 1.