Q: What is the prime factorization of the number 209,559,168?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 13 x 1,999
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 7, 13, 1,999 }
  • Written in exponential form: 27 x 32 x 71 x 131 x 1,9991

Why is the prime factorization of 209,559,168 written as 27 x 32 x 71 x 131 x 1,9991?

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 209,559,168

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 209,559,168 by 2

209,559,168 ÷ 2 = 104,779,584 - No remainder! 2 is one of the factors!
104,779,584 ÷ 2 = 52,389,792 - No remainder! 2 is one of the factors!
52,389,792 ÷ 2 = 26,194,896 - No remainder! 2 is one of the factors!
26,194,896 ÷ 2 = 13,097,448 - No remainder! 2 is one of the factors!
13,097,448 ÷ 2 = 6,548,724 - No remainder! 2 is one of the factors!
6,548,724 ÷ 2 = 3,274,362 - No remainder! 2 is one of the factors!
3,274,362 ÷ 2 = 1,637,181 - No remainder! 2 is one of the factors!
1,637,181 ÷ 2 = 818,590.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,637,181 ÷ 3 = 545,727 - No remainder! 3 is one of the factors!
545,727 ÷ 3 = 181,909 - No remainder! 3 is one of the factors!
181,909 ÷ 3 = 60,636.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
181,909 ÷ 5 = 36,381.8 - This has a remainder. 5 is not a factor.
181,909 ÷ 7 = 25,987 - No remainder! 7 is one of the factors!
25,987 ÷ 7 = 3,712.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
25,987 ÷ 11 = 2,362.4545 - This has a remainder. 11 is not a factor.
25,987 ÷ 13 = 1,999 - No remainder! 13 is one of the factors!
1,999 ÷ 13 = 153.7692 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
1,999 ÷ 17 = 117.5882 - This has a remainder. 17 is not a factor.
1,999 ÷ 19 = 105.2105 - This has a remainder. 19 is not a factor.
1,999 ÷ 23 = 86.913 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,999 ÷ 1,999 = 1 - No remainder! 1,999 is one of the factors!

The orange divisor(s) above are the prime factors of the number 209,559,168. 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 3 x 3 x 7 x 13 x 1,999 = 209,559,168. It can also be written in exponential form as 27 x 32 x 71 x 131 x 1,9991.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 209,559,168.

209,559,168
Factor Arrows
2104,779,584
Factor Arrows
252,389,792
Factor Arrows
226,194,896
Factor Arrows
213,097,448
Factor Arrows
26,548,724
Factor Arrows
23,274,362
Factor Arrows
21,637,181
Factor Arrows
3545,727
Factor Arrows
3181,909
Factor Arrows
725,987
Factor Arrows
131,999

More Prime Factorization Examples

209,559,166209,559,167209,559,169209,559,170
21 x 104,779,5831431 x 2931 x 16,63312,3111 x 90,679121 x 51 x 171 x 191 x 64,8791

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