Is 931 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 that is 1.
   
• As 931 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 931 has more than two factors, it is not a prime number. Few 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 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 931 is prime or composite.

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

Step 2: Divide 931 by 2. It is not divisible by 2, so 2 is not a factor of 931. 

Step 3: Divide 931 by 3. It is not divisible by 3, so 3 is not a factor of 931.

Step 4: You can simplify checking divisors up to 931 by finding the square root value. We then need to only check divisors up to the square root value (approximately 30.5).

Since 931 has factors other than 1 and itself, 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 odd. Therefore, 931 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 931 is 13. Since 13 is not divisible by 3, 931 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 1. Therefore, 931 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 (93 - 2 = 91). Since 91 is divisible by 7, 931 is also divisible by 7.

Divisibility by 11: In 931, 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.

Since 931 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 numbers in a range, e.g., 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 have marked all prime numbers.

Through this process, we create a list of prime numbers. 931 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 931 as 7 × 133, where 7 is a prime number. 

Step 2: In 133, further break it down to 7 × 19, where 19 is a prime number.

Hence, the prime factorization of 931 is 7 × 7 × 19.

• 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.
   
• Twin primes: The pair of prime numbers whose difference is 2 are called twin prime numbers. For example, (11, 13) is a pair of twin primes because the difference of 13 and 11 is 2.
   
• 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.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.
   
• Divisibility: The ability of one number to be divided by another without leaving a remainder. For example, 10 is divisible by 2, 5, and 10.