Is 317 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 317 has exactly two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 317 has exactly two factors, it is a prime number. A few methods are used to distinguish between prime and composite numbers. These 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 317 is prime or composite.

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

Step 2: Check divisibility of 317 by 2, 3, 5, 7, 11, 13, 17, etc. up to the square root of 317, which is approximately 17.8.

Step 3: 317 is not divisible by any of these numbers.

Since 317 has only 2 divisors (1 and 317), 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: 317 is an odd number, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in 317 is 11, which is not divisible by 3.

Divisibility by 5: The unit’s place digit is 7, so 317 is not divisible by 5.

Divisibility by 7: Performing the divisibility test for 7, we see that 317 is not divisible by 7.

Divisibility by 11: In 317, the alternating sum of the digits is 5, which is not divisible by 11.

Since 317 is not divisible by any of these, it confirms that 317 is a prime 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 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 have marked and crossed all numbers up to the desired limit.

Since 317 is not crossed out in this list, it confirms that 317 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: Attempt to divide 317 by the smallest prime numbers such as 2, 3, 5, 7, 11, 13, and 17.

Step 2: Since 317 is not divisible by any of these, it cannot be broken down into smaller prime factors.

Hence, 317 itself is a prime number.

• Prime Numbers: Natural numbers greater than 1 with only two factors, 1 and the number itself.
   
• Composite Numbers: Natural numbers greater than 1 with more than two factors.
   
• Divisibility Test: A set of rules used to determine if one number is divisible by another.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.
   
• Prime Factorization: Expressing a number as the product of its prime factors.