Q: What is the prime factorization of the number 100,025,344?

 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 23 x 31 x 137
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 23, 31, 137 }
  • Written in exponential form: 210 x 231 x 311 x 1371

Why is the prime factorization of 100,025,344 written as 210 x 231 x 311 x 1371?

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 100,025,344

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 100,025,344 by 2

100,025,344 ÷ 2 = 50,012,672 - No remainder! 2 is one of the factors!
50,012,672 ÷ 2 = 25,006,336 - No remainder! 2 is one of the factors!
25,006,336 ÷ 2 = 12,503,168 - No remainder! 2 is one of the factors!
12,503,168 ÷ 2 = 6,251,584 - No remainder! 2 is one of the factors!
6,251,584 ÷ 2 = 3,125,792 - No remainder! 2 is one of the factors!
3,125,792 ÷ 2 = 1,562,896 - No remainder! 2 is one of the factors!
1,562,896 ÷ 2 = 781,448 - No remainder! 2 is one of the factors!
781,448 ÷ 2 = 390,724 - No remainder! 2 is one of the factors!
390,724 ÷ 2 = 195,362 - No remainder! 2 is one of the factors!
195,362 ÷ 2 = 97,681 - No remainder! 2 is one of the factors!
97,681 ÷ 2 = 48,840.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
97,681 ÷ 3 = 32,560.3333 - This has a remainder. 3 is not a factor.
97,681 ÷ 5 = 19,536.2 - This has a remainder. 5 is not a factor.
97,681 ÷ 7 = 13,954.4286 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
97,681 ÷ 23 = 4,247 - No remainder! 23 is one of the factors!
4,247 ÷ 23 = 184.6522 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
4,247 ÷ 29 = 146.4483 - This has a remainder. 29 is not a factor.
4,247 ÷ 31 = 137 - No remainder! 31 is one of the factors!
137 ÷ 31 = 4.4194 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
137 ÷ 37 = 3.7027 - This has a remainder. 37 is not a factor.
137 ÷ 41 = 3.3415 - This has a remainder. 41 is not a factor.
137 ÷ 43 = 3.186 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
137 ÷ 137 = 1 - No remainder! 137 is one of the factors!

The orange divisor(s) above are the prime factors of the number 100,025,344. 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 23 x 31 x 137 = 100,025,344. It can also be written in exponential form as 210 x 231 x 311 x 1371.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 100,025,344.

100,025,344
Factor Arrows
250,012,672
Factor Arrows
225,006,336
Factor Arrows
212,503,168
Factor Arrows
26,251,584
Factor Arrows
23,125,792
Factor Arrows
21,562,896
Factor Arrows
2781,448
Factor Arrows
2390,724
Factor Arrows
2195,362
Factor Arrows
297,681
Factor Arrows
234,247
Factor Arrows
31137

More Prime Factorization Examples

100,025,342100,025,343100,025,345100,025,346
21 x 50,012,671132 x 111 x 1,010,357151 x 71 x 2,857,867121 x 31 x 2,0811 x 8,0111

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