Q: What is the prime factorization of the number 233,321,809?

 A:
  • The prime factors are: 7 x 277 x 120,331
    • or also written as { 7, 277, 120,331 }
  • Written in exponential form: 71 x 2771 x 120,3311

Why is the prime factorization of 233,321,809 written as 71 x 2771 x 120,3311?

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 233,321,809

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 233,321,809 by 2

233,321,809 ÷ 2 = 116,660,904.5 - This has a remainder. Let's try another prime number.
233,321,809 ÷ 3 = 77,773,936.3333 - This has a remainder. Let's try another prime number.
233,321,809 ÷ 5 = 46,664,361.8 - This has a remainder. Let's try another prime number.
233,321,809 ÷ 7 = 33,331,687 - No remainder! 7 is one of the factors!
33,331,687 ÷ 7 = 4,761,669.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
33,331,687 ÷ 11 = 3,030,153.3636 - This has a remainder. 11 is not a factor.
33,331,687 ÷ 13 = 2,563,975.9231 - This has a remainder. 13 is not a factor.
33,331,687 ÷ 17 = 1,960,687.4706 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
33,331,687 ÷ 277 = 120,331 - No remainder! 277 is one of the factors!
120,331 ÷ 277 = 434.4079 - There is a remainder. We can't divide by 277 evenly anymore. Let's try the next prime number
120,331 ÷ 281 = 428.2242 - This has a remainder. 281 is not a factor.
120,331 ÷ 283 = 425.1979 - This has a remainder. 283 is not a factor.
120,331 ÷ 293 = 410.686 - This has a remainder. 293 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
120,331 ÷ 120,331 = 1 - No remainder! 120,331 is one of the factors!

The orange divisor(s) above are the prime factors of the number 233,321,809. If we put all of it together we have the factors 7 x 277 x 120,331 = 233,321,809. It can also be written in exponential form as 71 x 2771 x 120,3311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 233,321,809.

233,321,809
Factor Arrows
733,331,687
Factor Arrows
277120,331

More Prime Factorization Examples

233,321,807233,321,808233,321,810233,321,811
4571 x 510,551124 x 31 x 4,860,871121 x 51 x 311 x 752,651131 x 3891 x 199,9331

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