Is 1036 a Prime Number?

Numbers can be classified into two types based on the number of factors they have:

prime numbers and composite numbers.

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 only 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 have several properties:

• Prime numbers are positive numbers 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.
   
• Since 1036 has more than two factors, it is not a prime number.
  

The defining characteristic of a prime number is that it has only two divisors: 1 and itself. Since 1036 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers, including:

• 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 divisors, we classify numbers as follows: - If there are exactly 2 divisors, the number is prime. - If the count is more than 2, the number is composite. Let’s check whether 1036 is prime or composite.

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

Step 2: Divide 1036 by 2. It is divisible by 2, so 2 is a factor of 1036.

Step 3: Divide 1036 by 3. It is not divisible by 3, so 3 is not a factor of 1036.

Step 4: To simplify checking divisors, find the square root of 1036, which is approximately 32. We then need to check divisors only up to this root value.

1036 has more than 2 divisors, indicating it is a composite number.

The divisibility test method involves using a set of rules to check whether a number is divisible by another number completely. Here are some examples:

Divisibility by 2: The last digit of 1036 is 6, an even number, so 1036 is divisible by 2.

Divisibility by 3: The sum of the digits in 1036 is 10, which is not divisible by 3, so 1036 is not divisible by 3. 

Divisibility by 4: The last two digits of 1036 are 36, which is divisible by 4.

Divisibility by 5: The last digit is not 0 or 5, so 1036 is not divisible by 5.

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

Since 1036 is divisible by more than two numbers, it is a composite number.

The prime number chart is created using a method called “The Sieve of Eratosthenes.” This method involves the following steps:

Step 1: Write numbers from 1 to 100 in 10 rows and 10 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: Continue this process until all numbers are either marked or crossed out, except 1. The chart reveals that 1036, being greater than 100, can be checked for primality by the absence of smaller factors.

Since 1036 is divisible by smaller prime numbers, it is not a prime number.

Prime factorization involves breaking down a number into prime factors and multiplying those factors to obtain the original number.

Step 1: We can write 1036 as 2 × 518.

Step 2: Break down 518 further into 2 × 259.

Step 3: Since 259 is divisible by 7 and 37, its prime factorization is 7 × 37.

Step 4: Therefore, the prime factorization of 1036 is 2 × 2 × 7 × 37.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers. For example, 12 is a composite number because it is divisible by 1, 2, 3, 4, 6, and 12.
   
• Prime numbers: Numbers greater than 1 that have no divisors other than 1 and itself. For example, 13 is a prime number.
   
• Divisibility rules: Rules used to determine whether one number is divisible by another.
   
• Prime factorization: Expressing a number as a product of its prime factors.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.