Q: What is the prime factorization of the number 813,456?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7 x 269
    • or also written as { 2, 2, 2, 2, 3, 3, 3, 7, 269 }
  • Written in exponential form: 24 x 33 x 71 x 2691

Why is the prime factorization of 813,456 written as 24 x 33 x 71 x 2691?

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 813,456

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 813,456 by 2

813,456 ÷ 2 = 406,728 - No remainder! 2 is one of the factors!
406,728 ÷ 2 = 203,364 - No remainder! 2 is one of the factors!
203,364 ÷ 2 = 101,682 - No remainder! 2 is one of the factors!
101,682 ÷ 2 = 50,841 - No remainder! 2 is one of the factors!
50,841 ÷ 2 = 25,420.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
50,841 ÷ 3 = 16,947 - No remainder! 3 is one of the factors!
16,947 ÷ 3 = 5,649 - No remainder! 3 is one of the factors!
5,649 ÷ 3 = 1,883 - No remainder! 3 is one of the factors!
1,883 ÷ 3 = 627.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,883 ÷ 5 = 376.6 - This has a remainder. 5 is not a factor.
1,883 ÷ 7 = 269 - No remainder! 7 is one of the factors!
269 ÷ 7 = 38.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
269 ÷ 11 = 24.4545 - This has a remainder. 11 is not a factor.
269 ÷ 13 = 20.6923 - This has a remainder. 13 is not a factor.
269 ÷ 17 = 15.8235 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
269 ÷ 269 = 1 - No remainder! 269 is one of the factors!

The orange divisor(s) above are the prime factors of the number 813,456. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7 x 269 = 813,456. It can also be written in exponential form as 24 x 33 x 71 x 2691.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 813,456.

813,456
Factor Arrows
2406,728
Factor Arrows
2203,364
Factor Arrows
2101,682
Factor Arrows
250,841
Factor Arrows
316,947
Factor Arrows
35,649
Factor Arrows
31,883
Factor Arrows
7269

More Prime Factorization Examples

813,454813,455813,457813,458
21 x 1011 x 4,027151 x 162,69111671 x 4,871121 x 406,7291

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