Is 568 a Prime Number?

There are two types of numbers, mostly —

Prime numbers and composite numbers, depending on the number of 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 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 follow few properties like:

• 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 that is 1.
   
• As 568 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 568 has more than two factors, it is not a prime number. Few methods are used to distinguish between prime and composite numbers. A few methods are:

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

The method in which we count the number of divisors to categorize the numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize prime and composite numbers. If there is a total count of only 2 divisors, then the number would be prime. If the count is more than 2, then the number is composite. Let’s check whether 568 is prime or composite.

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

Step 2: Divide 568 by 2. It is divisible by 2, so 2 is a factor of 568.

Step 3: Divide 568 by 3. It is not divisible by 3, so 3 is not a factor of 568.

Step 4: You can simplify checking divisors up to 568 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 568 by 2, 4, and 8, it is divisible by 2, 4, and 8.

Since 568 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 8. Eight is an even number, which means that 568 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 568 is 19. Since 19 is not divisible by 3, 568 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 8. Therefore, 568 is not divisible by 5.

Divisibility by 7: The last digit in 568 is 8. To check divisibility by 7, double the last digit (8 × 2 = 16). Then, subtract it from the rest of the number (56 - 16 = 40). Since 40 is not divisible by 7, 568 is also not divisible by 7.

Divisibility by 11: In 568, the alternating sum of the digits (5 - 6 + 8 = 7) is not divisible by 11. Hence, 568 is not divisible by 11.

Since 568 is divisible by numbers other than 1 and itself, it has more than two factors. Therefore, it is a composite number.

The prime number chart is a tool created by using a method called “The Sieve of Eratosthenes.” In this method, we follow the following steps.

Step 1: Write numbers in sequence up to a desired limit.

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 will have a list of prime numbers within the range.

Since 568 is not present on the list of prime numbers, it is a composite number.

Prime factorization is a process of breaking down a number into prime factors. Then multiply those factors to obtain the original number.

Step 1: We can write 568 as 2 × 284.

Step 2: In 2 × 284, 284 is a composite number. Further, break 284 into 2 × 142.

Step 3: Further, break 142 into 2 × 71.

Step 4: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 568 is 2 × 2 × 2 × 71 (or \(2^3 \times 71\)).

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and itself. For example, 5 is a prime number because it is only divisible by 1 and 5.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers. For example, 12 is a composite number because 12 is divisible by 1, 2, 3, 4, 6, and 12.

• Divisibility test: A set of rules to determine if one number is divisible by another without performing division.

• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 18 is \(2 \times 3 \times 3\) or \(2 \times 3^2\).

• Co-prime numbers: Two numbers are co-prime if their greatest common divisor (GCD) is 1. For example, 8 and 15 are co-prime numbers because they do not have any common factors other than 1.