Is 1269 a Prime Number?

There are mainly two types of numbers — 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 1269 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 1269 has more than two factors, it is not a prime number. Several 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 numbers as prime or composite.

- 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 1269 is prime or composite.

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

- Step 2: Divide 1269 by 2. It is not divisible by 2, so 2 is not a factor of 1269.

- Step 3: Divide 1269 by 3. It is divisible by 3, so 3 is a factor of 1269.

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

- Step 5: When we divide 1269 by other numbers up to its square root, we find additional factors such as 3 and 423.

Since 1269 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: 1269 is not even, so it is not divisible by 2.

- Divisibility by 3: The sum of the digits in the number 1269 is 18. Since 18 is divisible by 3, 1269 is also divisible by 3.

- Divisibility by 5: The unit’s place digit is 9. Therefore, 1269 is not divisible by 5.

- Divisibility by 7: To check divisibility by 7, double the last digit (9 × 2 = 18), then subtract it from the rest of the number (126 - 18 = 108). Since 108 is divisible by 7, 1269 is divisible by 7.

- Divisibility by 11: Alternating sum of digits is 1 - 2 + 6 - 9 = -4, which is not divisible by 11. Hence, 1269 is not divisible by 11.

Since 1269 has more than two factors, it is a composite number.

The prime number chart is a tool created by using a method called "The Sieve of Eratosthenes." In this method, we follow these steps:

- Step 1: Write numbers from 1 to 100 in a 10 × 10 grid.

- Step 2: Leave 1 unmarked as it is neither prime nor composite.

- Step 3: Mark 2 as it is a prime number and cross out all the multiples of 2.

- Step 4: Mark 3 as it is a prime number and cross out all the multiples of 3.

- Step 5: Repeat this process for subsequent numbers. Through this process, we will have a list of prime numbers.

Since 1269 is not on the list of primes, it is a composite number.

Prime factorization is a process of breaking down a number into its prime factors. Then, multiply those factors to obtain the original number.

- Step 1: We can write 1269 as 3 × 423.

- Step 2: In 3 × 423, 423 is a composite number. Further, break down 423 into 3 × 141.

- Step 3: Break down 141 further into 3 × 47 (since 47 is a prime number).

Hence, the prime factorization of 1269 is 3 × 3 × 3 × 47.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 12 is a composite number because it is divisible by 1, 2, 3, 4, 6, and 12.
  
   
• Divisibility rules: Simple rules that help determine whether one number is divisible by another without performing division.
  
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
  
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.
  
   
• Co-prime numbers: Two numbers are co-prime if their greatest common divisor is 1. For example, 8 and 15 are co-prime.