Q: What is the prime factorization of the number 813,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 11 x 11
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 5, 7, 11, 11 }
  • Written in exponential form: 26 x 31 x 51 x 71 x 112

Why is the prime factorization of 813,120 written as 26 x 31 x 51 x 71 x 112?

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 813,120

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 813,120 by 2

813,120 ÷ 2 = 406,560 - No remainder! 2 is one of the factors!
406,560 ÷ 2 = 203,280 - No remainder! 2 is one of the factors!
203,280 ÷ 2 = 101,640 - No remainder! 2 is one of the factors!
101,640 ÷ 2 = 50,820 - No remainder! 2 is one of the factors!
50,820 ÷ 2 = 25,410 - No remainder! 2 is one of the factors!
25,410 ÷ 2 = 12,705 - No remainder! 2 is one of the factors!
12,705 ÷ 2 = 6,352.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
12,705 ÷ 3 = 4,235 - No remainder! 3 is one of the factors!
4,235 ÷ 3 = 1,411.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
4,235 ÷ 5 = 847 - No remainder! 5 is one of the factors!
847 ÷ 5 = 169.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
847 ÷ 7 = 121 - No remainder! 7 is one of the factors!
121 ÷ 7 = 17.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
121 ÷ 11 = 11 - No remainder! 11 is one of the factors!
11 ÷ 11 = 1 - No remainder! 11 is one of the factors!

The orange divisor(s) above are the prime factors of the number 813,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 11 x 11 = 813,120. It can also be written in exponential form as 26 x 31 x 51 x 71 x 112.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 813,120.

813,120
Factor Arrows
2406,560
Factor Arrows
2203,280
Factor Arrows
2101,640
Factor Arrows
250,820
Factor Arrows
225,410
Factor Arrows
212,705
Factor Arrows
34,235
Factor Arrows
5847
Factor Arrows
7121
Factor Arrows
1111

More Prime Factorization Examples

813,118813,119813,121813,122
21 x 406,5591231 x 35,3531813,121121 x 406,5611

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