Is 357 a Prime Number?

There are two main types of numbers — prime numbers and composite numbers, depending on the number of factors they have. 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 such as: -

• 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 because they have only one common factor, which is 1.
• Since 357 has more than two factors, it is not a prime number.

The defining characteristic of a prime number is that it has only two divisors: 1 and itself. Since 357 has more than two factors, it is not a prime number. Several methods can be used to distinguish between prime and composite numbers. Some of these methods include: -

1. Counting Divisors Method 
2. Divisibility Test 
3. Prime Number Chart 
4. Prime Factorization

The counting divisors method involves counting the number of divisors a number has to categorize it as either prime or composite. -

• If there is a total count of only 2 divisors, then the number is prime. 
• If the count is more than 2, then the number is composite. Let’s check whether 357 is prime or composite.

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

Step 2: Divide 357 by 2. It is not divisible by 2, as it is an odd number.

Step 3: Divide 357 by 3. The sum of the digits (3 + 5 + 7 = 15) is divisible by 3, so 357 is divisible by 3.

Step 4: Continue checking divisors up to the square root of 357, approximately 18.9.

Since 357 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is completely divisible by another number, known as the Divisibility Test Method. -

Divisibility by 2: 357 is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits (3 + 5 + 7 = 15) is divisible by 3, so 357 is divisible by 3. 

Divisibility by 5: The last digit is 7, so 357 is not divisible by 5. 

Divisibility by 7: Doubling the last digit (7 × 2 = 14) and subtracting from the rest of the number (35 - 14 = 21), 21 is divisible by 7, so 357 is divisible by 7. 

Divisibility by 11: The difference between the sum of the digits in odd positions and even positions is (3 + 7) - 5 = 5, which is not divisible by 11.

Since 357 is divisible by 3 and 7, it has more than two factors and is therefore a composite number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” This method involves the following steps:

Step 1: Write numbers from 1 to 100 in 10 rows and 10 columns.

Step 2: Leave 1 unmarked, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all multiples of 3.

Step 5: Repeat this process until the table is filled with marked and crossed boxes, except for 1. Through this process, we will have a list of prime numbers from 1 to 100.

Since 357 is not in this list, and it has divisors other than 1 and itself, it is a composite number.

Prime factorization is a process of breaking down a number into its prime factors, then multiplying those factors to obtain the original number.

Step 1: We can write 357 as 3 × 119.

Step 2: In 3 × 119, 3 is a prime number. Further, break down 119 into 7 × 17, where both 7 and 17 are primes.

Step 3: The complete prime factorization of 357 is 3 × 7 × 17.

• 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 it is divisible by 1, 2, 3, 4, 6, and 12. 

• Divisibility rules: Guidelines used to 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 breaking down a number into its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7. 

• Co-prime numbers: Two numbers are co-prime if their greatest common divisor (GCD) is 1. For example, 8 and 15 are co-prime. 

• Natural numbers: A set of positive integers starting from 1 and increasing without bound.