Is 841 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 have a few properties: 

• 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 841 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 841 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers, such as: 

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

The method in which we count the number of divisors to categorize numbers as prime or composite is called the counting divisors method. Based on the count of divisors, we categorize numbers as prime or composite.

• If there is a total count of only 2 divisors, the number is prime.

• If the count is more than 2, the number is composite.

Let’s check whether 841 is prime or composite.

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

Step 2: Divide 841 by 2. It is not divisible by 2, so 2 is not a factor of 841.

Step 3: Continue testing divisibility by smaller prime numbers up to its square root (~29).

Step 4: 841 is divisible by 29 (since 29 × 29 = 841). Since 841 is divisible by numbers other than 1 and itself, it is a composite number.

Divisibility rules help check whether a number is divisible by another number completely. This is called the Divisibility Test Method.

While 841 is not divisible by 2, 3, or 5, we found it is divisible by 29.

Since 841 can be divided by 29, it has more than two factors, making it a composite number.

The prime number chart is a tool created using “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 coloring or crossing, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all multiples of 3.

Step 5: Continue this process up to the square root of 100. Using this chart, identify prime numbers up to 100, confirming that 841 is not among them.

Therefore, it is a composite number.

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

Step 1: Start with 841.

Step 2: Since 841 = 29 × 29, and 29 is a prime number, we have the prime factorization.

Step 3: The prime factorization of 841 is 29 × 29.

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

• Perfect square: A number that is the square of an integer. For example, 49 is a perfect square because it is 7 × 7. 

• Prime numbers: Natural numbers greater than 1 with exactly two divisors, 1 and itself. For example, 5 is a prime number. 

• Divisibility rules: Guidelines to determine if one number is divisible by another without performing division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.

• Factors: Numbers that divide another number exactly without leaving a remainder. For example, the factors of 8 are 1, 2, 4, and 8.