Is 521 a Prime Number?

Numbers can be categorized as prime numbers or composite numbers based on their 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 integer that has more than two factors.

For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have several properties, such as:

• Prime numbers are positive integers 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.

Since 521 has only two factors, it is a prime number.

A prime number is characterized by having exactly two divisors: 1 and itself. Since 521 has only two factors, it is a prime number.

There are several methods to determine whether a number is prime or composite, such as:

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

The counting divisors method involves determining the number of divisors a number has to classify it as prime or composite.

• If there are exactly 2 divisors, then the number is prime.
   
• If there are more than 2 divisors, then the number is composite. Let’s check whether 521 is prime or composite.

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

Step 2: Divide 521 by numbers up to its square root (approximately 22.8). It is not divisible by any other numbers.

Since 521 has exactly two divisors, it is a prime number.

The divisibility test method involves using rules to check whether a number is divisible by another number without leaving a remainder.

Divisibility by 2: 521 is odd, so it is not divisible by 2.

Divisibility by 3: The sum of the digits in 521 is 8, which is not divisible by 3.

Divisibility by 5: 521 does not end in 0 or 5, so it is not divisible by 5.

Divisibility by 7: Applying the rule for 7 shows that 521 is not divisible by 7.

Divisibility by 11: Alternating sums do not give a difference divisible by 11.

Since 521 is not divisible by any number other than 1 and 521 itself, it is a prime number.

The prime number chart is a tool created using "The Sieve of Eratosthenes." This method includes:

Step 1: Writing numbers from 1 to 1000 in rows and columns.

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

Step 3: Marking 2 as prime and crossing out all its multiples.

Step 4: Continuing this process for the next unmarked number. Through this process, you'll find that 521 is a prime number since it does not appear as a multiple of any smaller prime number.

Prime factorization involves breaking down a number into its prime factors. Once broken down, you multiply these factors to reconstruct the original number.

Step 1: Check divisibility by prime numbers smaller than the square root of 521.

Step 2: If no divisors are found other than 1 and 521 itself, the number is prime.

Since 521 cannot be broken down into smaller prime factors, it is a prime number.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves.
   
• Composite numbers: Natural numbers greater than 1 that have more than two divisors.
   
• Divisibility Test: A method used to determine if one number is divisible by another without a remainder.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.