Is 1293 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 that is 1.

As 1293 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 1293 has more than two factors, it is not 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 1293 is prime or composite.

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

Step 2: Divide 1293 by 2. It is not divisible by 2, so 2 is not a factor.

Step 3: Divide 1293 by 3. It is divisible by 3, so 3 is a factor of 1293.

Step 4: You can simplify checking divisors up to 1293 by finding the square root value. We then need to only check divisors up to the root value.

Step 5: When we divide 1293 by 3, 431, and others, it is divisible by 3 and 431.

Since 1293 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. This is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 3. Since 3 is not an even number, 1293 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 1293 is 15. Since 15 is divisible by 3, 1293 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 3. Therefore, 1293 is not divisible by 5.

Divisibility by 7: The last digit in 1293 is 3. To check divisibility by 7, double the last digit (3 × 2 = 6) and subtract it from the rest of the number (129 - 6 = 123). Since 123 is divisible by 7, 1293 is also divisible by 7.

Divisibility by 11: In 1293, the sum of the digits in odd positions is 12, and the sum of the digits in even positions is 9. The difference is 3, which is not divisible by 11. Since 1293 is divisible by numbers other than 1 and itself, it has more than two factors.

Therefore, it is a composite 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 from 1 to 1299 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 reach the table consisting of marked and crossed boxes, except 1. Through this process, you will have a list of prime numbers up to 1299.

1293 is not present in the list of prime numbers, so it is a composite number.

Prime factorization is a process of breaking down a number into prime factors and then multiplying those factors to obtain the original number.

Step 1: We can write 1293 as 3 × 431.

Step 2: 3 is a prime number, and 431 is also prime.

Hence, the prime factorization of 1293 is 3 × 431.

• Composite Numbers: Natural numbers greater than 1 that have more than 2 factors. For example, 12 is a composite number because 12 is divisible by 1, 2, 3, 4, 6, and 12.

• Prime Numbers: Natural numbers greater than 1 with exactly two distinct positive divisors: 1 and the number itself. For example, 5 is a prime number.

• Divisibility: A number is divisible by another if it can be divided by that number without leaving a remainder.

• Prime Factorization: Breaking down a number into its prime number components. For example, the prime factorization of 18 is 2 × 3 × 3.

• Co-prime Numbers: Two numbers are co-prime if their greatest common divisor is 1. For example, 8 and 15 are co-prime.