Is 1209 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 that is 1.
• As 1209 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 1209 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers: -

1. Counting Divisors Method 
2. Divisibility Test 
3. Prime Number Chart 
4. 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 1209 is prime or composite. 

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

Step 2: Divide 1209 by 2. It is not divisible by 2, so 2 is not a factor of 1209. 

Step 3: Divide 1209 by 3. It is divisible by 3, so 3 is a factor of 1209. 

Step 4: You can simplify checking divisors up to 1209 by finding the square root. We only need to check divisors up to the square root. 

Step 5: When we divide 1209 by 3, 13, and 31, it is divisible by 3, 13, and 31.

Since 1209 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 1209 is odd, so it is not divisible by 2. -

Divisibility by 3: The sum of the digits in the number 1209 is 12. Since 12 is divisible by 3, 1209 is also divisible by 3. -

Divisibility by 5: The unit’s place digit is 9. Therefore, 1209 is not divisible by 5. -

Divisibility by 7: The last digit in 1209 is 9. To check divisibility by 7, double the last digit (9 × 2 = 18). Then, subtract it from the rest of the number (120 - 18 = 102). Since 102 is not divisible by 7, 1209 is also not divisible by 7. -

Divisibility by 11: In 1209, the difference between the sum of the digits in odd positions (1+0) and even positions (2+9) is 8. This would mean that 1209 is not divisible by 11.

Since 1209 is divisible by 3, 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 these steps: 

Step 1: Write 1 to 100 in 10 rows and 10 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 reach the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100.

The list 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. 1209 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 and then multiplying those factors to obtain the original number. -

Step 1: We can write 1209 as 3 × 403. 

Step 2: In 3 × 403, 403 is a composite number. Further, break the 403 into 13 × 31. 

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

Hence, the prime factorization of 1209 is 3 × 13 × 31.

• 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.

• Prime factorization: The process of breaking down a composite number into its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Divisibility rules: Guidelines to determine if one number is divisible by another without performing division.

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

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.