Is 1297 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 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 1297 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 1297 has exactly two factors, it is a prime number. A few methods are used to distinguish between prime and composite numbers:

• 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 1297 is prime or composite.

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

Step 2: Check divisibility by numbers up to the square root of 1297.

Step 3: The square root of 1297 is approximately 36, so we check divisibility up to 36.

Step 4: 1297 is not divisible by any number from 2 to 36.

Since 1297 has only 1 and itself as 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: 1297 is not even, so it is not divisible by 2. -

Divisibility by 3: The sum of the digits (1 + 2 + 9 + 7 = 19) is not divisible by 3.

Divisibility by 5: The unit’s place digit is not 0 or 5, so 1297 is not divisible by 5.

Divisibility by 7: Applying divisibility rules, 1297 is not divisible by 7.

Divisibility by 11: Applying divisibility rules, 1297 is not divisible by 11. Since 1297 is not divisible by any number other than 1 and itself, 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 numbers in a sequence up to a desired limit.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark prime numbers and cross out all their multiples.

Step 4: Continue this process until the desired range is covered.

Since 1297 is not crossed out in this method, 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: Begin with the smallest prime number and check if it divides 1297.

Step 2: Continue with the next smallest prime numbers.

Step 3: Since no prime numbers up to the square root of 1297 divide it evenly, 1297 is a prime number itself.

• 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 2 numbers. 

• Divisibility: The ability of one number to be divided by another without leaving a remainder.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.

• Prime factorization: The process of breaking down a number into its prime factors.