Q: What is the prime factorization of the number 120,513,024?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 26,153
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 26,153 }
  • Written in exponential form: 29 x 32 x 26,1531

Why is the prime factorization of 120,513,024 written as 29 x 32 x 26,1531?

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 120,513,024

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 120,513,024 by 2

120,513,024 ÷ 2 = 60,256,512 - No remainder! 2 is one of the factors!
60,256,512 ÷ 2 = 30,128,256 - No remainder! 2 is one of the factors!
30,128,256 ÷ 2 = 15,064,128 - No remainder! 2 is one of the factors!
15,064,128 ÷ 2 = 7,532,064 - No remainder! 2 is one of the factors!
7,532,064 ÷ 2 = 3,766,032 - No remainder! 2 is one of the factors!
3,766,032 ÷ 2 = 1,883,016 - No remainder! 2 is one of the factors!
1,883,016 ÷ 2 = 941,508 - No remainder! 2 is one of the factors!
941,508 ÷ 2 = 470,754 - No remainder! 2 is one of the factors!
470,754 ÷ 2 = 235,377 - No remainder! 2 is one of the factors!
235,377 ÷ 2 = 117,688.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
235,377 ÷ 3 = 78,459 - No remainder! 3 is one of the factors!
78,459 ÷ 3 = 26,153 - No remainder! 3 is one of the factors!
26,153 ÷ 3 = 8,717.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
26,153 ÷ 5 = 5,230.6 - This has a remainder. 5 is not a factor.
26,153 ÷ 7 = 3,736.1429 - This has a remainder. 7 is not a factor.
26,153 ÷ 11 = 2,377.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
26,153 ÷ 26,153 = 1 - No remainder! 26,153 is one of the factors!

The orange divisor(s) above are the prime factors of the number 120,513,024. 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 3 x 3 x 26,153 = 120,513,024. It can also be written in exponential form as 29 x 32 x 26,1531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 120,513,024.

120,513,024
Factor Arrows
260,256,512
Factor Arrows
230,128,256
Factor Arrows
215,064,128
Factor Arrows
27,532,064
Factor Arrows
23,766,032
Factor Arrows
21,883,016
Factor Arrows
2941,508
Factor Arrows
2470,754
Factor Arrows
2235,377
Factor Arrows
378,459
Factor Arrows
326,153

More Prime Factorization Examples

120,513,022120,513,023120,513,025120,513,026
21 x 71 x 411 x 209,9531120,513,023152 x 1,7211 x 2,801121 x 60,256,5131

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