Is 398 a Prime Number?

There are two main types of numbers — 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, such as:

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 398 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 398 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers:

• 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 numbers as prime or composite.

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 398 is prime or composite.

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

Step 2: Divide 398 by 2. It is divisible by 2, so 2 is a factor of 398.

Step 3: Divide 398 by 3. It is not divisible by 3, so 3 is not a factor of 398.

Step 4: You can simplify checking divisors up to 398 by finding the square root value. We then need to check divisors only up to the square root value.

Step 5: When we divide 398 by 2, 199, and other numbers, it is divisible by 2 and 199. Since 398 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. This is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 8, which is even, so 398 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 398 is 20. Since 20 is not divisible by 3, 398 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 8, so 398 is not divisible by 5.

Divisibility by 7: The last digit in 398 is 8. To check divisibility by 7, double the last digit (8 × 2 = 16). Then, subtract it from the rest of the number (39 - 16 = 23). Since 23 is not divisible by 7, 398 is also not divisible by 7.

Divisibility by 11: In 398, the sum of the digits in odd positions is 11, and the sum of the digits in even positions is 9. The difference is 2, so 398 is not divisible by 11. Since 398 is divisible by numbers like 2 and 199, it has more than two factors.

Therefore, it is a composite number.

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

Step 1: Write 1 to 100 in 10 rows and 10 columns.

Step 2: Leave 1 without marking, 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 includes 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. 398 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 multiplying those factors to obtain the original number.

Step 1: We can write 398 as 2 × 199.

Step 2: Both 2 and 199 are prime numbers.

Hence, the prime factorization of 398 is 2 × 199.

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

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and itself.

• Factors: The numbers that divide a given number exactly without leaving a remainder. For example, the factors of 4 are 1, 2, and 4.

• Divisibility rules: A set of rules that help determine whether a 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.