Is 251 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 251 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 251 has only two factors, it is a prime number. A 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 251 is prime or composite. -

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

Step 2: Divide 251 by 2. It is not divisible by 2, so 2 is not a factor of 251. 

Step 3: Continue dividing 251 by successive numbers up to the square root of 251 (approximately 15.8). 

Step 4: 251 is not divisible by any of these numbers.

Since 251 has only 2 divisors, it is a prime 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 1. Since 1 is not an even number, 251 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 251 is 8. Since 8 is not divisible by 3, 251 is also not divisible by 3. - Divisibility by 5: The unit’s place digit is 1. Therefore, 251 is not divisible by 5. 

Divisibility by 7: Double the last digit (1 × 2 = 2) and subtract it from the rest of the number (25 - 2 = 23). Since 23 is not divisible by 7, 251 is also not divisible by 7. 

Divisibility by 11: The difference between the sum of digits in odd positions and the sum of digits in even positions is 1, which is not divisible by 11.

Since 251 is not divisible by any of these numbers, it is a prime 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.

Since 251 is greater than 100, we continue the pattern mentally or use other methods to determine its primality. 251 is not listed as a composite in any extended chart; therefore, it is a prime 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: Check divisibility by prime numbers such as 2, 3, 5, 7, 11, and up to the square root of 251. 

Step 2: Since 251 is not divisible by any of these prime numbers, it cannot be broken down further.

Thus, 251 is only divisible by 1 and 251 itself.

This confirms that 251 is a prime number.

• Prime Number: A natural number greater than 1 that is not divisible by any other numbers except 1 and itself. 
   
• Composite Number: A natural number greater than 1 that is not a prime number, meaning it has more than two factors.
   
• Divisibility Rules: Guidelines that help determine if one number can be divided 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.