Q: What is the prime factorization of the number 230,438,400?

 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 5 x 5 x 17 x 353
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353 }
  • Written in exponential form: 29 x 31 x 52 x 171 x 3531

Why is the prime factorization of 230,438,400 written as 29 x 31 x 52 x 171 x 3531?

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 230,438,400

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 230,438,400 by 2

230,438,400 ÷ 2 = 115,219,200 - No remainder! 2 is one of the factors!
115,219,200 ÷ 2 = 57,609,600 - No remainder! 2 is one of the factors!
57,609,600 ÷ 2 = 28,804,800 - No remainder! 2 is one of the factors!
28,804,800 ÷ 2 = 14,402,400 - No remainder! 2 is one of the factors!
14,402,400 ÷ 2 = 7,201,200 - No remainder! 2 is one of the factors!
7,201,200 ÷ 2 = 3,600,600 - No remainder! 2 is one of the factors!
3,600,600 ÷ 2 = 1,800,300 - No remainder! 2 is one of the factors!
1,800,300 ÷ 2 = 900,150 - No remainder! 2 is one of the factors!
900,150 ÷ 2 = 450,075 - No remainder! 2 is one of the factors!
450,075 ÷ 2 = 225,037.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
450,075 ÷ 3 = 150,025 - No remainder! 3 is one of the factors!
150,025 ÷ 3 = 50,008.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
150,025 ÷ 5 = 30,005 - No remainder! 5 is one of the factors!
30,005 ÷ 5 = 6,001 - No remainder! 5 is one of the factors!
6,001 ÷ 5 = 1,200.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
6,001 ÷ 7 = 857.2857 - This has a remainder. 7 is not a factor.
6,001 ÷ 11 = 545.5455 - This has a remainder. 11 is not a factor.
6,001 ÷ 13 = 461.6154 - This has a remainder. 13 is not a factor.
6,001 ÷ 17 = 353 - No remainder! 17 is one of the factors!
353 ÷ 17 = 20.7647 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
353 ÷ 19 = 18.5789 - This has a remainder. 19 is not a factor.
353 ÷ 23 = 15.3478 - This has a remainder. 23 is not a factor.
353 ÷ 29 = 12.1724 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
353 ÷ 353 = 1 - No remainder! 353 is one of the factors!

The orange divisor(s) above are the prime factors of the number 230,438,400. 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 2 x 3 x 5 x 5 x 17 x 353 = 230,438,400. It can also be written in exponential form as 29 x 31 x 52 x 171 x 3531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 230,438,400.

230,438,400
Factor Arrows
2115,219,200
Factor Arrows
257,609,600
Factor Arrows
228,804,800
Factor Arrows
214,402,400
Factor Arrows
27,201,200
Factor Arrows
23,600,600
Factor Arrows
21,800,300
Factor Arrows
2900,150
Factor Arrows
2450,075
Factor Arrows
3150,025
Factor Arrows
530,005
Factor Arrows
56,001
Factor Arrows
17353

More Prime Factorization Examples

230,438,398230,438,399230,438,401230,438,402
21 x 1031 x 2331 x 4,80116,5711 x 35,06919,6771 x 23,813121 x 115,219,2011

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