Is 1357 a Prime Number?

There are two primary types of numbers based on the number of factors they possess:

prime numbers and composite numbers.

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.

Composite numbers are positive numbers that are divisible by more than two numbers. For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have specific 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.

As 1357 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 1357 has more than two factors, it is not a prime number. Several methods can be used to distinguish between prime and composite numbers, including:

• Counting Divisors Method
  
   
• Divisibility Test
  
   
• Prime Number Chart
  
   
• Prime Factorization

The counting divisors method involves counting the number of divisors to categorize numbers as prime or composite. Based on the number of divisors, a number can be classified as:

- Prime if there is a total count of only 2 divisors.

- Composite if the count is more than 2.

Let’s check whether 1357 is prime or composite.

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

Step 2: Divide 1357 by 2. It is not divisible by 2, so 2 is not a factor of 1357.

Step 3: Divide 1357 by 3, 5, 7, and so on, up to the square root of 1357, which is approximately 36.8.

Through this process, we find that 1357 is divisible by 19 and 71.

Since 1357 has more than 2 divisors, it is a composite number.

The Divisibility Test Method uses a set of rules to check whether a number is divisible by another number without a remainder. Here’s how it applies to 1357:

- Divisibility by 2: 1357 is odd, so it is not divisible by 2.

- Divisibility by 3: The sum of the digits (1 + 3 + 5 + 7 = 16) is not divisible by 3, so 1357 is not divisible by 3.

- Divisibility by 5: 1357 does not end in 0 or 5, so it is not divisible by 5.

- Divisibility by 7: Using a divisibility rule for 7 or direct division shows it is not divisible by 7.

- Divisibility by 11: Alternating sum of digits (1 - 3 + 5 - 7 = -4) is not divisible by 11, so 1357 is not divisible by 11

- Divisibility by 19: 1357 divided by 19 equals 71, so it is divisible by 19. Since 1357 is divisible by 19, it has more than two factors, thus 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 1000 in 10 rows and 100 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 identify prime numbers.

Since 1357 is not in the list of prime numbers, it is a composite number.

Prime factorization involves breaking down a number into its prime factors and then multiplying them to obtain the original number.

Step 1: Start with the smallest prime number, 19, and divide 1357 by 19 to get 71.

Step 2: Since 71 is a prime number, the prime factorization of 1357 is 19 × 71.

• Prime Numbers: Natural numbers greater than 1 that are divisible only by 1 and themselves. For example, 7 is a prime number.
  
   
• Composite Numbers: Natural numbers greater than 1 that have more than two distinct positive divisors. For example, 12 is a composite number.
  
   
• Divisibility: A property that determines if one number can be divided by another without leaving a remainder.
  
   
• Prime Factorization: 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 is 1. For example, 8 and 15 are co-prime.