Is 1026 a Prime Number?

Numbers can generally be categorized into two types—

prime numbers and composite numbers—based on their factors.

A prime number is a natural number that is divisible only by 1 and itself.

For instance, 3 is a prime number because it is divisible by only 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 such as:

• 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 1026 has more than two factors, it is not a prime number.
  

A prime number is characterized by having only two divisors: 1 and itself. Since 1026 has more than two factors, it is not a prime number. Several methods can be used to distinguish between prime and composite numbers:

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

The counting divisors method involves counting the number of divisors to categorize numbers as prime or composite. Based on the count of the divisors, we can classify numbers as follows: If there is a total count of only 2 divisors, then the number is prime. If the count is more than 2, then the number is composite. Let’s check whether 1026 is prime or composite.

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

Step 2: Divide 1026 by 2. It is divisible by 2, so 2 is a factor of 1026.

Step 3: Divide 1026 by 3. It is divisible by 3, so 3 is a factor of 1026.

Step 4: Simplify checking divisors up to the square root of 1026. We then need to check divisors only up to this root value.

Step 5: When we divide 1026 by 2, 3, and other numbers up to its square root, we find multiple divisors.

Since 1026 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. This is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 6, which is even, indicating that 1026 is divisible by 2.

Divisibility by 3: The sum of the digits of 1026 is 9 (1+0+2+6), which is divisible by 3, so 1026 is divisible by 3.

Divisibility by 5: The unit’s place digit is 6, so 1026 is not divisible by 5.

Divisibility by 7: Using divisibility rules for 7, we find that 1026 is divisible by 7.

Divisibility by 11: The alternating sum of the digits of 1026 is 3 (1-0+2-6), which is not divisible by 11.

Since 1026 is divisible by 2, 3, and 7, it has more than two factors and is therefore 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 1000 in rows and columns.

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

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

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

Step 5: Repeat this process up to the square root of the highest number.

Through this process, we obtain a list of prime numbers.

Since 1026 is not in this list of prime numbers, it is a composite number.

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

Step 1: We can write 1026 as 2 × 513.

Step 2: In 2 × 513, 513 is a composite number. Further, break 513 into its prime factors: 513 = 3 × 171.

Step 3: Further breaking down 171, we get 171 = 3 × 57.

Step 4: Finally, 57 can be broken down into 3 × 19.

The prime factorization of 1026 is 2 × 3 × 3 × 3 × 19.

• Prime numbers: Natural numbers greater than 1 that are divisible by only 1 and itself are called prime numbers. For example, 5 is a prime number because it is only divisible by 1 and 5.
  
   
• 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.
  
   
• Factors: The numbers that divide a number exactly without leaving a remainder are called factors. For example, the factors of 9 are 1, 3, and 9.
  
   
• Divisibility rules: Rules that help determine whether a number is divisible by another without performing full division. For example, if the sum of a number's digits is divisible by 3, the number itself is divisible by 3.
  
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.