Q: What is the prime factorization of the number 810,216?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 3 x 3 x 11 x 11 x 31
    • or also written as { 2, 2, 2, 3, 3, 3, 11, 11, 31 }
  • Written in exponential form: 23 x 33 x 112 x 311

Why is the prime factorization of 810,216 written as 23 x 33 x 112 x 311?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 810,216

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 810,216 by 2

810,216 ÷ 2 = 405,108 - No remainder! 2 is one of the factors!
405,108 ÷ 2 = 202,554 - No remainder! 2 is one of the factors!
202,554 ÷ 2 = 101,277 - No remainder! 2 is one of the factors!
101,277 ÷ 2 = 50,638.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
101,277 ÷ 3 = 33,759 - No remainder! 3 is one of the factors!
33,759 ÷ 3 = 11,253 - No remainder! 3 is one of the factors!
11,253 ÷ 3 = 3,751 - No remainder! 3 is one of the factors!
3,751 ÷ 3 = 1,250.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,751 ÷ 5 = 750.2 - This has a remainder. 5 is not a factor.
3,751 ÷ 7 = 535.8571 - This has a remainder. 7 is not a factor.
3,751 ÷ 11 = 341 - No remainder! 11 is one of the factors!
341 ÷ 11 = 31 - No remainder! 11 is one of the factors!
31 ÷ 11 = 2.8182 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
31 ÷ 13 = 2.3846 - This has a remainder. 13 is not a factor.
31 ÷ 17 = 1.8235 - This has a remainder. 17 is not a factor.
31 ÷ 19 = 1.6316 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
31 ÷ 31 = 1 - No remainder! 31 is one of the factors!

The orange divisor(s) above are the prime factors of the number 810,216. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 3 x 3 x 11 x 11 x 31 = 810,216. It can also be written in exponential form as 23 x 33 x 112 x 311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 810,216.

810,216
Factor Arrows
2405,108
Factor Arrows
2202,554
Factor Arrows
2101,277
Factor Arrows
333,759
Factor Arrows
311,253
Factor Arrows
33,751
Factor Arrows
11341
Factor Arrows
1131

More Prime Factorization Examples

810,214810,215810,217810,218
21 x 1931 x 2,099151 x 72 x 3,3071191 x 42,643121 x 1371 x 2,9571

Try the factor calculator

Explore more about the number 810,216:


Ask a Question