Is 1197 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.
• Since 1197 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 1197 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. These methods include: -

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 1197 is prime or composite.

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

Step 2: Divide 1197 by 2. It is not divisible by 2, so 2 is not a factor of 1197.

Step 3: Divide 1197 by 3. It is divisible by 3, so 3 is a factor of 1197.

Step 4: You can simplify checking divisors up to 1197 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 1197 by 3, 7, and 13, it is divisible by these numbers.

Since 1197 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: The number 1197 is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 1197 is 18. Since 18 is divisible by 3, 1197 is also divisible by 3. 

Divisibility by 5: The unit’s place digit is 7. Therefore, 1197 is not divisible by 5. 

Divisibility by 7: The last digit in 1197 is 7. To check divisibility by 7, double the last digit (7 × 2 = 14). Then, subtract it from the rest of the number (119 - 14 = 105). Since 105 is divisible by 7, 1197 is divisible by 7. -

Divisibility by 11: In 1197, the sum of the digits in odd positions is 10, and the sum of the digits in even positions is 9. The difference between these sums is 1, which is not divisible by 11.

Since 1197 is divisible by numbers other than 1 and itself, 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 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. 1197 is not present in this 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 1197 as 3 × 399.

Step 2: In 3 × 399, 399 is a composite number. Further, break 399 into 3 × 133.

Step 3: Now further factor 133 into 7 × 19.

Hence, the prime factorization of 1197 is 3 × 3 × 7 × 19.

Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers. For example, 1197 is a composite number because it is divisible by 1, 3, 7, 19, 21, 57, 133, 399, and 1197. -

Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, and 7. 

Divisibility rules: Guidelines 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. 

Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 1197 is 3 × 3 × 7 × 19. 

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