Is 209 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, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers follow a 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 209 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 209 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 209 is prime or composite.

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

Step 2: Divide 209 by 2. It is not divisible by 2, so 2 is not a factor of 209.

Step 3: Divide 209 by 3. It is not divisible by 3, so 3 is not a factor of 209.

Step 4: You can simplify checking divisors up to 209 by finding the root value. We then need to only check divisors up to the root value.

Step 5: Divide 209 by 11. It is divisible by 11, so 11 is a factor of 209.

Since 209 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. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value of 209 is 9, which is odd, meaning 209 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 209 is 11. Since 11 is not divisible by 3, 209 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 9. Therefore, 209 is not divisible by 5.

Divisibility by 7: Subtract twice the last digit from the rest of the number (20 - 18 = 2). Since 2 is not divisible by 7, 209 is also not divisible by 7.

Divisibility by 11: The difference between the sum of the digits in odd positions (9) and even positions (2) is 7, which is not divisible by 11. Therefore, 209 is divisible by 11.

Since 209 is divisible by 11, it has more than two factors. Therefore, it is 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 sequence, covering a reasonable range.

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 reach the range of interest.

Through this process, we will have a list of prime numbers. 209 is not present in the list of prime numbers, so 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 209 as 11 × 19.

Step 2: Both 11 and 19 are prime numbers.

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

Hence, the prime factorization of 209 is 11 × 19.

• Prime numbers: Natural numbers greater than 1, having only two factors, 1 and itself. For example, 5 is a prime number because it can only be divided 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 12 is divisible by 1, 2, 3, 4, 6, and 12.
   
• Divisibility: The condition in which one integer can be divided by another without a remainder.
   
• Co-prime numbers: Two numbers that have only 1 as their common factor. For example, 15 and 28 are co-prime.
   
• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.