Q: What is the prime factorization of the number 320,212,233?

 A:
  • The prime factors are: 3 x 3 x 11 x 23 x 140,629
    • or also written as { 3, 3, 11, 23, 140,629 }
  • Written in exponential form: 32 x 111 x 231 x 140,6291

Why is the prime factorization of 320,212,233 written as 32 x 111 x 231 x 140,6291?

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 320,212,233

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 320,212,233 by 2

320,212,233 ÷ 2 = 160,106,116.5 - This has a remainder. Let's try another prime number.
320,212,233 ÷ 3 = 106,737,411 - No remainder! 3 is one of the factors!
106,737,411 ÷ 3 = 35,579,137 - No remainder! 3 is one of the factors!
35,579,137 ÷ 3 = 11,859,712.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
35,579,137 ÷ 5 = 7,115,827.4 - This has a remainder. 5 is not a factor.
35,579,137 ÷ 7 = 5,082,733.8571 - This has a remainder. 7 is not a factor.
35,579,137 ÷ 11 = 3,234,467 - No remainder! 11 is one of the factors!
3,234,467 ÷ 11 = 294,042.4545 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
3,234,467 ÷ 13 = 248,805.1538 - This has a remainder. 13 is not a factor.
3,234,467 ÷ 17 = 190,262.7647 - This has a remainder. 17 is not a factor.
3,234,467 ÷ 19 = 170,235.1053 - This has a remainder. 19 is not a factor.
3,234,467 ÷ 23 = 140,629 - No remainder! 23 is one of the factors!
140,629 ÷ 23 = 6,114.3043 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
140,629 ÷ 29 = 4,849.2759 - This has a remainder. 29 is not a factor.
140,629 ÷ 31 = 4,536.4194 - This has a remainder. 31 is not a factor.
140,629 ÷ 37 = 3,800.7838 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
140,629 ÷ 140,629 = 1 - No remainder! 140,629 is one of the factors!

The orange divisor(s) above are the prime factors of the number 320,212,233. If we put all of it together we have the factors 3 x 3 x 11 x 23 x 140,629 = 320,212,233. It can also be written in exponential form as 32 x 111 x 231 x 140,6291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 320,212,233.

320,212,233
Factor Arrows
3106,737,411
Factor Arrows
335,579,137
Factor Arrows
113,234,467
Factor Arrows
23140,629

More Prime Factorization Examples

320,212,231320,212,232320,212,234320,212,235
2631 x 1,217,537123 x 5711 x 70,099121 x 591 x 2571 x 10,559151 x 71 x 4091 x 22,3691

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