Is 1354 a Prime Number?

There are two main types of numbers based on the number of factors they have: 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 only 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 possess several 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 numbers because they have only one common factor, which is 1.
  
   
• As 1354 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 1354 has more than two factors, it is not a prime number. Several methods can be used to determine whether a number is prime or composite, including:

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

The counting divisors method involves counting the number of divisors a number has to determine whether it is prime or composite.

- If there are only 2 divisors, the number is prime.

- If there are more than 2 divisors, the number is composite. Let’s check whether 1354 is prime or composite.

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

Step 2: Divide 1354 by 2. It is divisible by 2, so 2 is a factor of 1354.

Step 3: Divide 1354 by 3. It is not divisible by 3, so 3 is not a factor.

Step 4: Divisors need to be checked only up to the square root of 1354.

Step 5: When we divide 1354 by 2, 677, and other divisors, it is divisible by 2, and 677.

Since 1354 has more than 2 divisors, it is a composite number.

The divisibility test method uses rules to determine whether a number is divisible by another number without a remainder.

- Divisibility by 2: The last digit of 1354 is 4, which is even, so 1354 is divisible by 2.

- Divisibility by 3: The sum of the digits is 13, which is not divisible by 3, so 1354 is not divisible by 3.

- Divisibility by 5: The last digit is not 0 or 5, so 1354 is not divisible by 5.

- Divisibility by 7: Using the divisibility rule for 7, 1354 is not divisible by 7.

- Divisibility by 11: The alternating sum of the digits (1 - 3 + 5 - 4 = -1) is not divisible by 11, so 1354 is not divisible by 11.

Since 1354 is divisible by 2, it has more than two factors, making it a composite number.

A 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 as a prime number and cross out all multiples of 2.

Step 4: Mark 3 as a prime number and cross out all multiples of 3.

Step 5: Continue this process until the table contains only marked and crossed boxes, except for 1. Through this process, a list of prime numbers up to 100 is obtained.

Since 1354 is not in this list and has divisors other than 1 and itself, it is a composite number.

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

Step 1: Start by dividing 1354 by the smallest prime number, 2, to get 677.

Step 2: 677 is a prime number, so the factorization ends here.

Step 3: The prime factorization of 1354 is 2 × 677.

• Composite numbers: Natural numbers greater than 1 that have more than two divisors. For example, 12 is a composite number because it is divisible by 1, 2, 3, 4, 6, and 12.
  
   
• Prime factorization: Breaking down a number into its smallest prime factors. For example, the prime factorization of 28 is 2 × 2 × 7.
  
   
• Divisibility rules: Guidelines that help determine if one number can be divided by another without a remainder. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.
  
   
• Sieve of Eratosthenes: A method for finding all prime numbers up to a certain limit by iteratively marking the multiples of each prime number starting from 2.
  
   
• Co-prime numbers: Two numbers that have only 1 as their common factor. For example, 8 and 15 are co-prime because their greatest common factor is 1.