Q: What is the prime factorization of the number 20,020,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 5 x 23 x 47 x 463
    • or also written as { 2, 2, 2, 5, 23, 47, 463 }
  • Written in exponential form: 23 x 51 x 231 x 471 x 4631

Why is the prime factorization of 20,020,120 written as 23 x 51 x 231 x 471 x 4631?

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 20,020,120

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 20,020,120 by 2

20,020,120 ÷ 2 = 10,010,060 - No remainder! 2 is one of the factors!
10,010,060 ÷ 2 = 5,005,030 - No remainder! 2 is one of the factors!
5,005,030 ÷ 2 = 2,502,515 - No remainder! 2 is one of the factors!
2,502,515 ÷ 2 = 1,251,257.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,502,515 ÷ 3 = 834,171.6667 - This has a remainder. 3 is not a factor.
2,502,515 ÷ 5 = 500,503 - No remainder! 5 is one of the factors!
500,503 ÷ 5 = 100,100.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
500,503 ÷ 7 = 71,500.4286 - This has a remainder. 7 is not a factor.
500,503 ÷ 11 = 45,500.2727 - This has a remainder. 11 is not a factor.
500,503 ÷ 13 = 38,500.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
500,503 ÷ 23 = 21,761 - No remainder! 23 is one of the factors!
21,761 ÷ 23 = 946.1304 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
21,761 ÷ 29 = 750.3793 - This has a remainder. 29 is not a factor.
21,761 ÷ 31 = 701.9677 - This has a remainder. 31 is not a factor.
21,761 ÷ 37 = 588.1351 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
21,761 ÷ 47 = 463 - No remainder! 47 is one of the factors!
463 ÷ 47 = 9.8511 - There is a remainder. We can't divide by 47 evenly anymore. Let's try the next prime number
463 ÷ 53 = 8.7358 - This has a remainder. 53 is not a factor.
463 ÷ 59 = 7.8475 - This has a remainder. 59 is not a factor.
463 ÷ 61 = 7.5902 - This has a remainder. 61 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
463 ÷ 463 = 1 - No remainder! 463 is one of the factors!

The orange divisor(s) above are the prime factors of the number 20,020,120. If we put all of it together we have the factors 2 x 2 x 2 x 5 x 23 x 47 x 463 = 20,020,120. It can also be written in exponential form as 23 x 51 x 231 x 471 x 4631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 20,020,120.

20,020,120
Factor Arrows
210,010,060
Factor Arrows
25,005,030
Factor Arrows
22,502,515
Factor Arrows
5500,503
Factor Arrows
2321,761
Factor Arrows
47463

More Prime Factorization Examples

20,020,11820,020,11920,020,12120,020,122
21 x 171 x 588,827131 x 71 x 1011 x 9,4391111 x 291 x 971 x 647121 x 34 x 123,5811

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