Is 712 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, such as:

• 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 712 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 712 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 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 712 is prime or composite.

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

Step 2: Divide 712 by 2. It is divisible by 2, so 2 is a factor of 712.

Step 3: Divide 712 by 3. It is not divisible by 3, so 3 is not a factor of 712.

Step 4: You can simplify checking divisors up to 712 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 712 by 2, 4, 8, and more, it is divisible by 2 and 4.

Since 712 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. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 2. Since 2 is an even number, 712 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 712 is 10. Since 10 is not divisible by 3, 712 is also not divisible by 3. -

Divisibility by 5: The unit’s place digit is 2. Therefore, 712 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (2 × 2 = 4). Then, subtract it from the rest of the number (71 - 4 = 67). Since 67 is not divisible by 7, 712 is also not divisible by 7. -

Divisibility by 11: In 712, the sum of the digits in odd positions is 3, and the sum of the digits in even positions is 1. This would mean that 712 is not divisible by 11.

Since 712 is divisible by more than two numbers, 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 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.

Since 712 is not present in the list of prime numbers, 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 712 as 2 × 356.

Step 2: In 356, break it further as 2 × 178.

Step 3: Then, 178 can be divided into 2 × 89.

Step 4: Finally, 89 is a prime number.

Hence, the prime factorization of 712 is 2 × 2 × 2 × 89.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than two numbers are called composite numbers. For example, 712 is a composite number because it is divisible by 1, 2, 4, 8, 89, 178, 356, and 712.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves are called prime numbers.

• Divisibility rules: Guidelines or methods to determine if one 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 a specified integer.