Q: What is the prime factorization of the number 226,666,323?

 A:
  • The prime factors are: 3 x 3 x 3 x 13 x 733 x 881
    • or also written as { 3, 3, 3, 13, 733, 881 }
  • Written in exponential form: 33 x 131 x 7331 x 8811

Why is the prime factorization of 226,666,323 written as 33 x 131 x 7331 x 8811?

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 226,666,323

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 226,666,323 by 2

226,666,323 ÷ 2 = 113,333,161.5 - This has a remainder. Let's try another prime number.
226,666,323 ÷ 3 = 75,555,441 - No remainder! 3 is one of the factors!
75,555,441 ÷ 3 = 25,185,147 - No remainder! 3 is one of the factors!
25,185,147 ÷ 3 = 8,395,049 - No remainder! 3 is one of the factors!
8,395,049 ÷ 3 = 2,798,349.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
8,395,049 ÷ 5 = 1,679,009.8 - This has a remainder. 5 is not a factor.
8,395,049 ÷ 7 = 1,199,292.7143 - This has a remainder. 7 is not a factor.
8,395,049 ÷ 11 = 763,186.2727 - This has a remainder. 11 is not a factor.
8,395,049 ÷ 13 = 645,773 - No remainder! 13 is one of the factors!
645,773 ÷ 13 = 49,674.8462 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
645,773 ÷ 17 = 37,986.6471 - This has a remainder. 17 is not a factor.
645,773 ÷ 19 = 33,988.0526 - This has a remainder. 19 is not a factor.
645,773 ÷ 23 = 28,077.087 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
645,773 ÷ 733 = 881 - No remainder! 733 is one of the factors!
881 ÷ 733 = 1.2019 - There is a remainder. We can't divide by 733 evenly anymore. Let's try the next prime number
881 ÷ 739 = 1.1922 - This has a remainder. 739 is not a factor.
881 ÷ 743 = 1.1857 - This has a remainder. 743 is not a factor.
881 ÷ 751 = 1.1731 - This has a remainder. 751 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
881 ÷ 881 = 1 - No remainder! 881 is one of the factors!

The orange divisor(s) above are the prime factors of the number 226,666,323. If we put all of it together we have the factors 3 x 3 x 3 x 13 x 733 x 881 = 226,666,323. It can also be written in exponential form as 33 x 131 x 7331 x 8811.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 226,666,323.

226,666,323
Factor Arrows
375,555,441
Factor Arrows
325,185,147
Factor Arrows
38,395,049
Factor Arrows
13645,773
Factor Arrows
733881

More Prime Factorization Examples

226,666,321226,666,322226,666,324226,666,325
71 x 171 x 1011 x 18,859121 x 113,333,161122 x 4611 x 122,921152 x 9,066,6531

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