Q: What is the prime factorization of the number 23,159,168?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 17 x 29 x 367
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 17, 29, 367 }
  • Written in exponential form: 27 x 171 x 291 x 3671

Why is the prime factorization of 23,159,168 written as 27 x 171 x 291 x 3671?

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 23,159,168

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 23,159,168 by 2

23,159,168 ÷ 2 = 11,579,584 - No remainder! 2 is one of the factors!
11,579,584 ÷ 2 = 5,789,792 - No remainder! 2 is one of the factors!
5,789,792 ÷ 2 = 2,894,896 - No remainder! 2 is one of the factors!
2,894,896 ÷ 2 = 1,447,448 - No remainder! 2 is one of the factors!
1,447,448 ÷ 2 = 723,724 - No remainder! 2 is one of the factors!
723,724 ÷ 2 = 361,862 - No remainder! 2 is one of the factors!
361,862 ÷ 2 = 180,931 - No remainder! 2 is one of the factors!
180,931 ÷ 2 = 90,465.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
180,931 ÷ 3 = 60,310.3333 - This has a remainder. 3 is not a factor.
180,931 ÷ 5 = 36,186.2 - This has a remainder. 5 is not a factor.
180,931 ÷ 7 = 25,847.2857 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
180,931 ÷ 17 = 10,643 - No remainder! 17 is one of the factors!
10,643 ÷ 17 = 626.0588 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
10,643 ÷ 19 = 560.1579 - This has a remainder. 19 is not a factor.
10,643 ÷ 23 = 462.7391 - This has a remainder. 23 is not a factor.
10,643 ÷ 29 = 367 - No remainder! 29 is one of the factors!
367 ÷ 29 = 12.6552 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
367 ÷ 31 = 11.8387 - This has a remainder. 31 is not a factor.
367 ÷ 37 = 9.9189 - This has a remainder. 37 is not a factor.
367 ÷ 41 = 8.9512 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
367 ÷ 367 = 1 - No remainder! 367 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,159,168. 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 17 x 29 x 367 = 23,159,168. It can also be written in exponential form as 27 x 171 x 291 x 3671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,159,168.

23,159,168
Factor Arrows
211,579,584
Factor Arrows
25,789,792
Factor Arrows
22,894,896
Factor Arrows
21,447,448
Factor Arrows
2723,724
Factor Arrows
2361,862
Factor Arrows
2180,931
Factor Arrows
1710,643
Factor Arrows
29367

More Prime Factorization Examples

23,159,16623,159,16723,159,16923,159,170
21 x 31 x 791 x 48,859123,159,167133 x 111 x 77,977121 x 51 x 311 x 74,7071

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