Is 545 a Prime Number?

Numbers can be categorized into two main types:

• prime numbers
   
• composite numbers

based on their factors. A prime number is a natural number greater than 1 that is divisible only by 1 and itself. For instance, 7 is a prime number because it is divisible by only 1 and 7. A composite number has more than two factors. For example, 12 is divisible by 1, 2, 3, 4, 6, and 12, making it a composite number. Properties of prime numbers include: 

Prime numbers are always greater than 1.

2 is the only even prime number.

They have exactly two distinct positive divisors: 1 and the number itself.

Any two distinct prime numbers are co-prime because their only common factor is 1. Since 545 has more than two factors, it is not a prime number.

The defining characteristic of a prime number is having exactly two divisors: 1 and itself. Since 545 has more than two factors, it is not a prime number. Several methods can be used to determine if a number is prime or composite:

• 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 if it is prime or composite. A number with exactly two divisors is prime, while a number with more than two divisors is composite. Let's examine whether 545 is prime or composite.

Step 1: Every number is divisible by 1 and itself.

Step 2: Divide 545 by 5. Since the last digit is 5, it is divisible by 5. 5 is a factor of 545.

Step 3: Divide 545 by 109. It is divisible by 109, so 109 is also a factor of 545.

Since 545 has more than two divisors (1, 5, 109, and 545), it is a composite number.

The divisibility test method involves a set of rules to determine if a number is divisible by another number without leaving a remainder.

Divisibility by 2: 545 is an odd number, so it is not divisible by 2.

Divisibility by 3: The sum of the digits (5 + 4 + 5 = 14) is not divisible by 3, so 545 is not divisible by 3.

Divisibility by 5: The last digit is 5, so 545 is divisible by 5.

Divisibility by 7: Double the last digit (5 × 2 = 10) and subtract from the rest (54 - 10 = 44). Since 44 is not divisible by 7, 545 is not divisible by 7.

Divisibility by 11: The difference between the sum of the digits in odd positions and even positions is 0, which is divisible by 11, so 545 is divisible by 11.

Since 545 is divisible by more numbers than just 1 and itself, it has more than two factors and is a composite number.

A prime number chart can be created using the Sieve of Eratosthenes to identify prime numbers up to a certain limit.

Step 1: Write numbers from 1 to 100 in a grid.

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

Step 3: Mark 2 as prime and cross out all its multiples.

Step 4: Mark 3 as prime and cross out all its multiples.

Step 5: Continue this process for the next numbers. Through this process, we get a list of prime numbers up to 100: 2, 3, 5, 7, 11, 13, 17, 19, etc.

Since 545 is not in this list and is divisible by 5 and 109, it is a composite number.

Prime factorization involves breaking down a composite number into its prime factors.

Step 1: Begin with 545 and find its smallest prime factor, which is 5.

Step 2: Divide 545 by 5 to get 109. Step 3: Since 109 is a prime number, we stop here. Thus, the prime factorization of 545 is 5 × 109.

• Composite numbers: Natural numbers greater than 1 that have more than two distinct positive divisors. For example, 10 is composite because it is divisible by 1, 2, 5, and 10.
   
• Divisibility rules: Guidelines that help determine whether one number is divisible by another without performing full division. For example, a number is divisible by 5 if its last digit is 0 or 5.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.
   
• Co-prime numbers: Two numbers that have no common factors other than 1. For example, 8 and 15 are co-prime.
   
• Sieve of Eratosthenes: An algorithm for finding all prime numbers up to a specified integer by iteratively marking the multiples of each prime starting from 2.