Is 311 a Prime Number?

There are two main 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 311 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 311 has exactly two factors, it is a prime number. Several methods are used to distinguish between prime and composite numbers. Some 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 311 is prime or composite.

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

Step 2: Divide 311 by 2. It is not divisible by 2, so 2 is not a factor of 311.

Step 3: Divide 311 by 3. It is not divisible by 3, so 3 is not a factor of 311.

Step 4: You can simplify checking divisors up to 311 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 311 by 2, 3, 5, 7, 11, 13, 17, etc., none of these are factors of 311.

Since 311 has only 2 divisors, it is a prime 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 one's place value is 1, which is odd, so 311 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 311 is 5. Since 5 is not divisible by 3, 311 is not divisible by 3.

Divisibility by 5: The unit's place digit is 1. Therefore, 311 is not divisible by 5.

Divisibility by 7: The last digit in 311 is 1. To check divisibility by 7, double the last digit (1 × 2 = 2). Then, subtract it from the rest of the number (31 - 2 = 29). Since 29 is not divisible by 7, 311 is not divisible by 7.

Divisibility by 11: In 311, the sum of the digits in odd positions is 4, and the sum of the digits in even positions is 1. This would mean that 311 is not divisible by 11.

Since 311 is not divisible by any of these numbers, it has only two factors, 1 and 311. Therefore, it is a prime 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 100 in 10 rows and 10 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 100.

Although 311 is beyond this list, using the same method up to 311 shows that it is a prime 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: Check if 311 can be divided by any prime numbers like 2, 3, 5, 7, 11, etc.

Step 2: 311 is not divisible by any of these prime numbers as shown in previous methods.

Step 3: Therefore, 311 can only be expressed as 1 × 311, indicating it is a prime number.

• Prime numbers: Natural numbers greater than 1 that are divisible by only 1 and themselves.
   
• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers.
   
• Divisibility test: A method to determine whether one number is divisible by another.
   
• Prime factorization: The process of expressing a number as a product of its prime factors.
   
• Co-prime numbers: Two numbers that have only one common factor, which is 1.