Is 1397 a Prime Number?

Numbers can primarily be classified as prime or composite based on their factors.

A prime number is a natural number greater than 1 that is only divisible by 1 and itself. For instance, 3 is a prime number because it is divisible by only 1 and 3.

A composite number has more than two factors. For example, 6 is a composite number because it is divisible by 1, 2, 3, and 6. Some properties of prime numbers include:  Prime numbers are greater than 1. 2 is the only even prime number. Prime numbers have only two factors: 1 and the number itself. Any two distinct prime numbers are co-prime, as they only share 1 as a common factor.

To determine if 1397 is a prime number, we must check its number of divisors.

A prime number has exactly two divisors: 1 and itself. To ascertain whether 1397 is a prime, we need to verify that it has no divisors other than 1 and 1397. Some methods to differentiate between prime and composite numbers include:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization

The counting divisors method involves counting all the divisors of a number to determine if it's prime or composite. If there are exactly 2 divisors, the number is prime. If there are more than 2 divisors, the number is composite. Let's verify if 1397 is prime:

Step 1: Every number is divisible by 1 and itself.

Step 2: Check divisibility from 2 up to the square root of 1397. Since 1397 is not divisible by any number other than 1 and itself, it is a prime number.

The divisibility test method involves using rules to check if a number is divisible by another number without a remainder.

Divisibility by 2: 1397 is odd, hence not divisible by 2.

Divisibility by 3: The sum of the digits (1 + 3 + 9 + 7 = 20) is not divisible by 3.

Divisibility by 5: The last digit is not 0 or 5, so it's not divisible by 5.

Further divisibility tests for other numbers up to the square root of 1397 confirm that 1397 is not divisible by any smaller prime number other than 1 and itself.

As 1397 is not divisible by any number other than 1 and 1397, it is indeed a prime number.

A prime number chart, often created using the Sieve of Eratosthenes method, helps identify prime numbers up to a certain limit.

Step 1: List numbers from 1 to 1000.

Step 2: Cross out multiples of each prime starting from 2.

Step 3: Continue this process up to the square root of the largest number in your list. In a detailed chart, 1397 would not have been crossed out, indicating its primality.

Prime factorization involves breaking down a number into its prime components.

Step 1: Attempt to divide 1397 by the smallest prime numbers.

Step 2: If no division yields a whole number other than 1 and 1397, it confirms that 1397 is a prime number. 1397 cannot be factorized into smaller prime numbers beyond 1 and itself, verifying its status as a prime number.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves.
   
• Composite numbers: Natural numbers greater than 1 that are divisible by more than two numbers.
   
• Divisibility: A number is divisible by another if the division results in an integer without a remainder.
   
• Prime factorization: Expressing a number as a product of its prime factors.
   
• Co-prime numbers: Two numbers that have no common factors other than 1.