Is 369 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 369 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 369 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers, including: -

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 369 is prime or composite:

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

Step 2: Divide 369 by 2. It is not divisible by 2, so 2 is not a factor of 369.

Step 3: Divide 369 by 3. It is divisible by 3, so 3 is a factor of 369.

Step 4: You can simplify checking divisors up to 369 by finding the root value. We then need to only check divisors up to the root value.

Since 369 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 is 9. Since 9 is an odd number, 369 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 369 is 18. Since 18 is divisible by 3, 369 is also divisible by 3. -

Divisibility by 5: The unit’s place digit is 9. Therefore, 369 is not divisible by 5. 

Divisibility by 7: To check divisibility by 7, double the last digit (9 × 2 = 18). Then, subtract it from the rest of the number (36 - 18 = 18). Since 18 is divisible by 7, 369 is also divisible by 7. 

Divisibility by 11: In 369, the sum of the digits in odd positions is 12, and the sum of the digits in even positions is 6. The difference is 6, which is not divisible by 11. Therefore, 369 is not divisible by 11.

Since 369 is divisible by 3 and 7, 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 up to a certain limit 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 reach the desired range. This process will give a list of prime numbers.

Since 369 is not present in the list of prime numbers, 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 369 as 3 × 123.

Step 2: In 3 × 123, 123 is a composite number. Further, break 123 into 3 × 41.

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

Hence, the prime factorization of 369 is 3 × 3 × 41.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 369 is a composite number because it is divisible by 1, 3, 9, 41, 123, and 369. 

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

• Prime Factorization: The process of breaking down a number into its smallest prime factors. For example, the prime factorization of 369 is 3 × 3 × 41. 

• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 369 are 1, 3, 9, 41, 123, and 369. 

• Sieve of Eratosthenes: A simple algorithm to find all prime numbers up to a specified integer. Through this method, numbers are systematically marked as prime or composite.