Q: What is the prime factorization of the number 522,720?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 11 x 11
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 3, 5, 11, 11 }
  • Written in exponential form: 25 x 33 x 51 x 112

Why is the prime factorization of 522,720 written as 25 x 33 x 51 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 522,720

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 522,720 by 2

522,720 ÷ 2 = 261,360 - No remainder! 2 is one of the factors!
261,360 ÷ 2 = 130,680 - No remainder! 2 is one of the factors!
130,680 ÷ 2 = 65,340 - No remainder! 2 is one of the factors!
65,340 ÷ 2 = 32,670 - No remainder! 2 is one of the factors!
32,670 ÷ 2 = 16,335 - No remainder! 2 is one of the factors!
16,335 ÷ 2 = 8,167.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
16,335 ÷ 3 = 5,445 - No remainder! 3 is one of the factors!
5,445 ÷ 3 = 1,815 - No remainder! 3 is one of the factors!
1,815 ÷ 3 = 605 - No remainder! 3 is one of the factors!
605 ÷ 3 = 201.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
605 ÷ 5 = 121 - No remainder! 5 is one of the factors!
121 ÷ 5 = 24.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
121 ÷ 7 = 17.2857 - This has a remainder. 7 is not a factor.
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 522,720. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 11 x 11 = 522,720. It can also be written in exponential form as 25 x 33 x 51 x 112.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 522,720.

522,720
Factor Arrows
2261,360
Factor Arrows
2130,680
Factor Arrows
265,340
Factor Arrows
232,670
Factor Arrows
216,335
Factor Arrows
35,445
Factor Arrows
31,815
Factor Arrows
3605
Factor Arrows
5121
Factor Arrows
1111

More Prime Factorization Examples

522,718522,719522,721522,722
21 x 71 x 37,3371522,7191231 x 22,727121 x 311 x 8,4311

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