Is 478 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 a 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, which is 1.

As 478 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 478 has more than two factors, it is not a prime number. 

• A few methods are used to distinguish between prime and composite numbers.

These methods include:

• 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 478 is prime or composite.
 

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

Step 2: Divide 478 by 2. It is divisible by 2, so 2 is a factor of 478.

Step 3: Divide 478 by 3. It is not divisible by 3, so 3 is not a factor of 478.
 

Step 4: You can simplify checking divisors up to 478 by finding the root value. We then need to only check divisors up to the root value. Step 5: When we divide 478 by 2, 239, and other numbers up to its square root, it is divisible by 2.

Since 478 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 one's place value is 8. Eight is an even number, which means that 478 is divisible by 2.
 

Divisibility by 3: The sum of the digits in the number 478 is 19. Since 19 is not divisible by 3, 478 is also not divisible by 3.
 

Divisibility by 5: The unit’s place digit is 8. Therefore, 478 is not divisible by 5.
 

Divisibility by 7: The last digit in 478 is 8. To check divisibility by 7, double the last digit (8 × 2 = 16). Then, subtract it from the rest of the number (47 - 16 = 31). Since 31 is not divisible by 7, 478 is also not divisible by 7.

Divisibility by 11: In 478, the sum of the digits in odd positions is 12, and the sum of the digits in even positions is 7. The difference is 5, which is not divisible by 11. Since 478 is divisible only by 2 and 239, 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 1 to 100 in 10 rows and 10 columns.

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 from 1 to 100.

The list is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

478 is not present in the list of prime numbers, so 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 478 as 2 × 239.

Step 2: Both 2 and 239 are prime numbers.

Therefore, the prime factorization of 478 is 2 × 239.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 478 is a composite number because it is divisible by 1, 2, 239, and 478.

• Prime numbers: Numbers greater than 1 that have no divisors other than 1 and themselves. For example, 3 and 5 are prime numbers.

• Divisibility Test: A method to determine if a number is divisible by another without performing full division.

• 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.