Q: What is the prime factorization of the number 204,230,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 11 x 967
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 11, 967 }
  • Written in exponential form: 28 x 31 x 52 x 111 x 9671

Why is the prime factorization of 204,230,400 written as 28 x 31 x 52 x 111 x 9671?

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 204,230,400

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 204,230,400 by 2

204,230,400 ÷ 2 = 102,115,200 - No remainder! 2 is one of the factors!
102,115,200 ÷ 2 = 51,057,600 - No remainder! 2 is one of the factors!
51,057,600 ÷ 2 = 25,528,800 - No remainder! 2 is one of the factors!
25,528,800 ÷ 2 = 12,764,400 - No remainder! 2 is one of the factors!
12,764,400 ÷ 2 = 6,382,200 - No remainder! 2 is one of the factors!
6,382,200 ÷ 2 = 3,191,100 - No remainder! 2 is one of the factors!
3,191,100 ÷ 2 = 1,595,550 - No remainder! 2 is one of the factors!
1,595,550 ÷ 2 = 797,775 - No remainder! 2 is one of the factors!
797,775 ÷ 2 = 398,887.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
797,775 ÷ 3 = 265,925 - No remainder! 3 is one of the factors!
265,925 ÷ 3 = 88,641.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
265,925 ÷ 5 = 53,185 - No remainder! 5 is one of the factors!
53,185 ÷ 5 = 10,637 - No remainder! 5 is one of the factors!
10,637 ÷ 5 = 2,127.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
10,637 ÷ 7 = 1,519.5714 - This has a remainder. 7 is not a factor.
10,637 ÷ 11 = 967 - No remainder! 11 is one of the factors!
967 ÷ 11 = 87.9091 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
967 ÷ 13 = 74.3846 - This has a remainder. 13 is not a factor.
967 ÷ 17 = 56.8824 - This has a remainder. 17 is not a factor.
967 ÷ 19 = 50.8947 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
967 ÷ 967 = 1 - No remainder! 967 is one of the factors!

The orange divisor(s) above are the prime factors of the number 204,230,400. 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 3 x 5 x 5 x 11 x 967 = 204,230,400. It can also be written in exponential form as 28 x 31 x 52 x 111 x 9671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 204,230,400.

204,230,400
Factor Arrows
2102,115,200
Factor Arrows
251,057,600
Factor Arrows
225,528,800
Factor Arrows
212,764,400
Factor Arrows
26,382,200
Factor Arrows
23,191,100
Factor Arrows
21,595,550
Factor Arrows
2797,775
Factor Arrows
3265,925
Factor Arrows
553,185
Factor Arrows
510,637
Factor Arrows
11967

More Prime Factorization Examples

204,230,398204,230,399204,230,401204,230,402
21 x 102,115,19913,3071 x 61,7571171 x 12,013,553121 x 8,8191 x 11,5791

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