Is 457 a Prime Number?

Numbers can generally be categorized into two types — prime numbers and composite numbers, based 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 only by 1 and 3.

A composite number is a positive number that has more than two factors. For instance, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have certain properties:

Prime numbers are positive numbers greater than 1.

2 is the only even prime number.

They have exactly 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 457 has only two factors, it is a prime number.

A key characteristic of a prime number is that it has exactly two divisors: 1 and itself. Since 457 meets this criterion, it is a prime number. Various methods can be used to identify prime numbers. Some of these methods include:

• Counting Divisors Method

• Divisibility Test

• Prime Number Chart

• Prime Factorization

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

Based on the count of the divisors, we can determine whether a number is prime or composite.

If there is a total count of exactly 2 divisors, then the number is prime.

If the count is more than 2, then the number is composite. Let’s check whether 457 is prime or composite.

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

Step 2: Check divisibility by numbers up to the square root of 457 (approximately 21.4). We only need to check divisibility by prime numbers up to this value.

Step 3: 457 is not divisible by 2, 3, 5, 7, 11, 13, 17, or 19. Since 457 is not divisible by any number other than 1 and itself, it is a prime number.

The divisibility test method uses a set of rules to check whether a number is divisible by another number completely. Let's apply these rules to 457:

Divisibility by 2: 457 is odd, so it is not divisible by 2.

Divisibility by 3: The sum of the digits (4 + 5 + 7 = 16) is not divisible by 3, so 457 is not divisible by 3.

Divisibility by 5: The last digit is not 0 or 5, so 457 is not divisible by 5.

Divisibility by 7, 11, 13, 17, and 19: Use respective divisibility rules or direct division to check divisibility. None divide 457 exactly. Since 457 is not divisible by any prime number other than itself, it is a prime number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” The steps involved are:

Step 1: Write numbers from 1 to 1000 in a grid.

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

Step 3: Mark 2, the smallest prime, and cross out all multiples of 2.

Step 4: Mark 3 and cross out all multiples of 3, and so on with subsequent primes. Continue this process to identify primes up to 1000. As 457 is not crossed out and is greater than 100 in the list, it is a prime number.

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

Step 1: Start with the smallest prime number, 2, and check if 457 is divisible by it. It is not.

Step 2: Proceed to the next prime numbers: 3, 5, 7, etc., up to the square root of 457 (about 21.4).

Step 3: 457 is not divisible by any of these primes. Since 457 cannot be broken down into smaller prime factors, it is itself 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 rules: Simple rules to determine if one number is divisible by another without performing division.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.

• Co-prime numbers: Two numbers are co-prime if their greatest common divisor is 1.