Is 507 a Prime Number?

There are 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 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 507 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 507 has more than two factors, it is not a prime number. Few methods are used to distinguish between prime and composite numbers. A few methods are: 

• 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 507 is prime or composite. 

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

Step 2: Divide 507 by 2. It is not divisible by 2, so 2 is not a factor of 507.

Step 3: Divide 507 by 3. The sum of the digits is 12, which is divisible by 3, so 3 is a factor of 507.

Step 4: You can simplify checking divisors up to 507 by finding the root value. We then need to only check divisors up to the root value.

Since 507 is divisible by 3, it has more than 2 divisors, making it a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method. 

Divisibility by 2: The number in the ones' place value is 7, an odd number, so 507 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 507 is 12, which is divisible by 3. Therefore, 507 is divisible by 3. 

Divisibility by 5: The unit’s place digit is 7. Therefore, 507 is not divisible by 5. 

Divisibility by 11: In 507, the alternating sum of the digits is 2, which is not divisible by 11.

Since 507 is divisible by 3, 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 the following steps:

Step 1: Write numbers in a range, such as 1 to 1000, in rows and columns. 

Step 2: Leave 1 without coloring or crossing, 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 until you have marked all prime numbers up to your desired range. Through this process, 507 will not be in the list of prime numbers, confirming it is a composite number.

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

Step 1: We can write 507 as 3 × 169. 

Step 2: In 3 × 169, 169 is a composite number. Further, break 169 into 13 × 13.

Step 3: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 507 is 3 × 13 × 13.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 507 is a composite number because it has factors such as 1, 3, 13, 39, 169, and 507. 
  
   
• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and itself. For example, 7 is a prime number. 
  
   
• Divisibility rules: Guidelines that help determine if one number is divisible by another without performing full division.
  
   
• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 507 is 3 × 13 × 13. 
  
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.