Is 1365 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, 12 is divisible by 1, 2, 3, 4, 6, and 12, making it a composite number.

Prime numbers follow a few properties:

- 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 1365 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 1365 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers:

• 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 1365 is prime or composite.

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

Step 2: Divide 1365 by 2. It is not divisible by 2, so 2 is not a factor.

Step 3: Divide 1365 by 3. It is divisible by 3, so 3 is a factor of 1365.

Step 4: Continue checking with other divisors. You can simplify this by checking divisors up to the square root of 1365.

Since 1365 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: 1365 is not an even number, so it is not divisible by 2.

- Divisibility by 3: The sum of the digits in 1365 is 15. Since 15 is divisible by 3, 1365 is divisible by 3.

- Divisibility by 5: The unit's place digit is 5. Therefore, 1365 is divisible by 5.

- Divisibility by 7: Double the last digit (5 × 2 = 10). Subtract it from the rest of the number (136 - 10 = 126). Since 126 is divisible by 7, 1365 is divisible by 7.

- Divisibility by 11: The alternating sum of the digits is 1 - 3 + 6 - 5 = -1, which is not divisible by 11. Since 1365 is divisible by 3, 5, and 7, 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 the following steps:

Step 1: Write numbers in a sequence.

Step 2: Leave 1 without marking, 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. Through this process, we will have a list of prime numbers.

Since 1365 is not on the list of prime numbers, it is a composite number.

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

Step 1: We can write 1365 as 3 × 455.

Step 2: Break down 455 into 5 × 91.

Step 3: Break down 91 into 7 × 13.

Step 4: Now we have the product consisting of only prime numbers.

Hence, the prime factorization of 1365 is 3 × 5 × 7 × 13.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 30 is a composite number because 30 is divisible by 1, 2, 3, 5, 6, 10, 15, and 30.
  
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.
  
   
• Divisibility: A number is divisible by another if the division results in a whole number without a remainder. For example, 10 is divisible by 5.
  
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to any given limit. It systematically marks the multiples of each prime number starting from 2.
  
   
• Co-prime numbers: Two numbers are co-prime if their greatest common divisor (GCD) is 1. For example, 8 and 15 are co-prime numbers.