Q: What is the prime factorization of the number 230,022,144?

 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 3 x 3 x 11 x 2,269
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 11, 2,269 }
  • Written in exponential form: 210 x 32 x 111 x 2,2691

Why is the prime factorization of 230,022,144 written as 210 x 32 x 111 x 2,2691?

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 230,022,144

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 230,022,144 by 2

230,022,144 ÷ 2 = 115,011,072 - No remainder! 2 is one of the factors!
115,011,072 ÷ 2 = 57,505,536 - No remainder! 2 is one of the factors!
57,505,536 ÷ 2 = 28,752,768 - No remainder! 2 is one of the factors!
28,752,768 ÷ 2 = 14,376,384 - No remainder! 2 is one of the factors!
14,376,384 ÷ 2 = 7,188,192 - No remainder! 2 is one of the factors!
7,188,192 ÷ 2 = 3,594,096 - No remainder! 2 is one of the factors!
3,594,096 ÷ 2 = 1,797,048 - No remainder! 2 is one of the factors!
1,797,048 ÷ 2 = 898,524 - No remainder! 2 is one of the factors!
898,524 ÷ 2 = 449,262 - No remainder! 2 is one of the factors!
449,262 ÷ 2 = 224,631 - No remainder! 2 is one of the factors!
224,631 ÷ 2 = 112,315.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
224,631 ÷ 3 = 74,877 - No remainder! 3 is one of the factors!
74,877 ÷ 3 = 24,959 - No remainder! 3 is one of the factors!
24,959 ÷ 3 = 8,319.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
24,959 ÷ 5 = 4,991.8 - This has a remainder. 5 is not a factor.
24,959 ÷ 7 = 3,565.5714 - This has a remainder. 7 is not a factor.
24,959 ÷ 11 = 2,269 - No remainder! 11 is one of the factors!
2,269 ÷ 11 = 206.2727 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
2,269 ÷ 13 = 174.5385 - This has a remainder. 13 is not a factor.
2,269 ÷ 17 = 133.4706 - This has a remainder. 17 is not a factor.
2,269 ÷ 19 = 119.4211 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,269 ÷ 2,269 = 1 - No remainder! 2,269 is one of the factors!

The orange divisor(s) above are the prime factors of the number 230,022,144. 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 3 x 3 x 11 x 2,269 = 230,022,144. It can also be written in exponential form as 210 x 32 x 111 x 2,2691.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 230,022,144.

230,022,144
Factor Arrows
2115,011,072
Factor Arrows
257,505,536
Factor Arrows
228,752,768
Factor Arrows
214,376,384
Factor Arrows
27,188,192
Factor Arrows
23,594,096
Factor Arrows
21,797,048
Factor Arrows
2898,524
Factor Arrows
2449,262
Factor Arrows
2224,631
Factor Arrows
374,877
Factor Arrows
324,959
Factor Arrows
112,269

More Prime Factorization Examples

230,022,142230,022,143230,022,145230,022,146
21 x 71 x 291 x 566,5571131 x 7611 x 23,251151 x 46,004,429121 x 1271 x 905,5991

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