Q: What is the prime factorization of the number 200,012,032?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 11 x 11 x 11 x 587
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 11, 11, 11, 587 }
  • Written in exponential form: 28 x 113 x 5871

Why is the prime factorization of 200,012,032 written as 28 x 113 x 5871?

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 200,012,032

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 200,012,032 by 2

200,012,032 ÷ 2 = 100,006,016 - No remainder! 2 is one of the factors!
100,006,016 ÷ 2 = 50,003,008 - No remainder! 2 is one of the factors!
50,003,008 ÷ 2 = 25,001,504 - No remainder! 2 is one of the factors!
25,001,504 ÷ 2 = 12,500,752 - No remainder! 2 is one of the factors!
12,500,752 ÷ 2 = 6,250,376 - No remainder! 2 is one of the factors!
6,250,376 ÷ 2 = 3,125,188 - No remainder! 2 is one of the factors!
3,125,188 ÷ 2 = 1,562,594 - No remainder! 2 is one of the factors!
1,562,594 ÷ 2 = 781,297 - No remainder! 2 is one of the factors!
781,297 ÷ 2 = 390,648.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
781,297 ÷ 3 = 260,432.3333 - This has a remainder. 3 is not a factor.
781,297 ÷ 5 = 156,259.4 - This has a remainder. 5 is not a factor.
781,297 ÷ 7 = 111,613.8571 - This has a remainder. 7 is not a factor.
781,297 ÷ 11 = 71,027 - No remainder! 11 is one of the factors!
71,027 ÷ 11 = 6,457 - No remainder! 11 is one of the factors!
6,457 ÷ 11 = 587 - No remainder! 11 is one of the factors!
587 ÷ 11 = 53.3636 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
587 ÷ 13 = 45.1538 - This has a remainder. 13 is not a factor.
587 ÷ 17 = 34.5294 - This has a remainder. 17 is not a factor.
587 ÷ 19 = 30.8947 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
587 ÷ 587 = 1 - No remainder! 587 is one of the factors!

The orange divisor(s) above are the prime factors of the number 200,012,032. 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 11 x 11 x 11 x 587 = 200,012,032. It can also be written in exponential form as 28 x 113 x 5871.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 200,012,032.

200,012,032
Factor Arrows
2100,006,016
Factor Arrows
250,003,008
Factor Arrows
225,001,504
Factor Arrows
212,500,752
Factor Arrows
26,250,376
Factor Arrows
23,125,188
Factor Arrows
21,562,594
Factor Arrows
2781,297
Factor Arrows
1171,027
Factor Arrows
116,457
Factor Arrows
11587

More Prime Factorization Examples

200,012,030200,012,031200,012,033200,012,034
21 x 51 x 20,001,203133 x 191 x 311 x 12,5771131 x 1511 x 101,891121 x 31 x 1131 x 1571 x 1,8791

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