Is 1301 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.

To determine if 1301 is a prime number, we need to establish if it has any factors other than 1 and itself.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 1301 does not have more than two factors, it is a prime number. Several methods can be used to determine whether a number is prime or composite, including:

• 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 1301 is prime or composite.

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

Step 2: Check divisibility by numbers up to the square root of 1301, which is approximately 36.05.

Step 3: Divisors to check are all primes up to 36: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. When we check 1301 against these divisors, it is not divisible by any of them.

Thus, 1301 has no divisors other than 1 and itself, confirming 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: 1301 is not even, so it is not divisible by 2. -

Divisibility by 3: The sum of the digits in 1301 is 5, which is not divisible by 3.

Divisibility by 5: The unit's place digit is 1, so 1301 is not divisible by 5.

- Other primes like 7, 11, 13, etc., up to the square root of 1301 are checked similarly. Since 1301 is not divisible by any of these primes, it 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 these steps:

Step 1: Write numbers up to a certain limit, say 1000, in rows and columns.

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

Step 3: Mark prime numbers and cross out all the multiples of each marked number.

Step 4: Continue marking and crossing out numbers until you have processed numbers up to the square root of your limit. 1301 is not in the range of the initial chart, but using the Sieve method,

we can find primes and verify that 1301 does not have divisors other than 1 and itself.

Prime factorization involves breaking down a number into its prime components, which are then multiplied to get the original number.

Step 1: Attempt to divide 1301 by the smallest prime numbers.

Step 2: 1301 is not divisible by any prime number less than its square root, as established previously.

Since no prime factors exist other than 1 and 1301 itself, 1301 is prime.

• 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 rules: Guidelines for determining whether one number can be divided by another without a remainder.

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

• Factors: Numbers that can be multiplied to produce another number. In terms of divisors, they are the numbers that divide a given number without leaving a remainder.