Is 585 a Prime Number?

Numbers can generally be categorized into two types —

Prime numbers and composite numbers, based on the number of factors they have.

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 have specific 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 585 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 585 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 method in which we count the number of divisors to categorize 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 585 is prime or composite.

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

Step 2: Divide 585 by 2. It is not divisible by 2, so 2 is not a factor.

Step 3: Divide 585 by 3. It is divisible by 3 (585 ÷ 3 = 195), so 3 is a factor.

Step 4: Continue checking divisors up to the square root of 585.

Step 5: When we divide 585 by 3, 5, 9, and 13, it is divisible by 3, 5, and 13.

Since 585 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, called the Divisibility Test Method. 

Divisibility by 2: The number 585 is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 585 is 18 (5 + 8 + 5), which is divisible by 3, so 585 is divisible by 3. 

Divisibility by 5: The unit’s place digit is 5, so 585 is divisible by 5. 

Divisibility by 7: Performing the divisibility test for 7 shows that 585 is not divisible by 7. 

Divisibility by 11: The alternating sum of the digits is 0 (5 - 8 + 5), which is divisible by 11, so 585 is divisible by 11.

Since 585 is divisible by more than just 1 and itself, it has more than two factors, making it 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 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: Repeat this process until you have marked all prime numbers up to 100.

The list of prime numbers from 1 to 100 is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. 585 is not present in the list of prime numbers, indicating it is a composite number.

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

Step 1: We can write 585 as 3 × 195.

Step 2: Break down 195 into 3 × 65.

Step 3: Break down 65 into 5 × 13.

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

Hence, the prime factorization of 585 is 3 × 3 × 5 × 13.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than two numbers are called composite numbers. For example, 12 is a composite number because it is divisible by 1, 2, 3, 4, 6, and 12.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 585 is 3 × 3 × 5 × 13.

• Divisibility rules: A set of rules that help determine if one number is divisible by another without performing division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.

• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4 × 4.

• Co-prime numbers: Two numbers that have no common factors other than 1. For example, 8 and 15 are co-prime.