Q: What is the prime factorization of the number 22,021,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 11 x 17 x 23
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 11, 17, 23 }
  • Written in exponential form: 210 x 51 x 111 x 171 x 231

Why is the prime factorization of 22,021,120 written as 210 x 51 x 111 x 171 x 231?

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 22,021,120

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 22,021,120 by 2

22,021,120 ÷ 2 = 11,010,560 - No remainder! 2 is one of the factors!
11,010,560 ÷ 2 = 5,505,280 - No remainder! 2 is one of the factors!
5,505,280 ÷ 2 = 2,752,640 - No remainder! 2 is one of the factors!
2,752,640 ÷ 2 = 1,376,320 - No remainder! 2 is one of the factors!
1,376,320 ÷ 2 = 688,160 - No remainder! 2 is one of the factors!
688,160 ÷ 2 = 344,080 - No remainder! 2 is one of the factors!
344,080 ÷ 2 = 172,040 - No remainder! 2 is one of the factors!
172,040 ÷ 2 = 86,020 - No remainder! 2 is one of the factors!
86,020 ÷ 2 = 43,010 - No remainder! 2 is one of the factors!
43,010 ÷ 2 = 21,505 - No remainder! 2 is one of the factors!
21,505 ÷ 2 = 10,752.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
21,505 ÷ 3 = 7,168.3333 - This has a remainder. 3 is not a factor.
21,505 ÷ 5 = 4,301 - No remainder! 5 is one of the factors!
4,301 ÷ 5 = 860.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
4,301 ÷ 7 = 614.4286 - This has a remainder. 7 is not a factor.
4,301 ÷ 11 = 391 - No remainder! 11 is one of the factors!
391 ÷ 11 = 35.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
391 ÷ 13 = 30.0769 - This has a remainder. 13 is not a factor.
391 ÷ 17 = 23 - No remainder! 17 is one of the factors!
23 ÷ 17 = 1.3529 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
23 ÷ 19 = 1.2105 - This has a remainder. 19 is not a factor.
23 ÷ 23 = 1 - No remainder! 23 is one of the factors!

The orange divisor(s) above are the prime factors of the number 22,021,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 11 x 17 x 23 = 22,021,120. It can also be written in exponential form as 210 x 51 x 111 x 171 x 231.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 22,021,120.

22,021,120
Factor Arrows
211,010,560
Factor Arrows
25,505,280
Factor Arrows
22,752,640
Factor Arrows
21,376,320
Factor Arrows
2688,160
Factor Arrows
2344,080
Factor Arrows
2172,040
Factor Arrows
286,020
Factor Arrows
243,010
Factor Arrows
221,505
Factor Arrows
54,301
Factor Arrows
11391
Factor Arrows
1723

More Prime Factorization Examples

22,021,11822,021,11922,021,12122,021,122
21 x 71 x 4391 x 3,583133 x 6531 x 1,2491291 x 1671 x 4,547121 x 31 x 611 x 60,1671

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