Q: What is the prime factorization of the number 82,809,600?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 19 x 227
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 19, 227 }
  • Written in exponential form: 28 x 31 x 52 x 191 x 2271

Why is the prime factorization of 82,809,600 written as 28 x 31 x 52 x 191 x 2271?

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 82,809,600

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 82,809,600 by 2

82,809,600 ÷ 2 = 41,404,800 - No remainder! 2 is one of the factors!
41,404,800 ÷ 2 = 20,702,400 - No remainder! 2 is one of the factors!
20,702,400 ÷ 2 = 10,351,200 - No remainder! 2 is one of the factors!
10,351,200 ÷ 2 = 5,175,600 - No remainder! 2 is one of the factors!
5,175,600 ÷ 2 = 2,587,800 - No remainder! 2 is one of the factors!
2,587,800 ÷ 2 = 1,293,900 - No remainder! 2 is one of the factors!
1,293,900 ÷ 2 = 646,950 - No remainder! 2 is one of the factors!
646,950 ÷ 2 = 323,475 - No remainder! 2 is one of the factors!
323,475 ÷ 2 = 161,737.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
323,475 ÷ 3 = 107,825 - No remainder! 3 is one of the factors!
107,825 ÷ 3 = 35,941.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
107,825 ÷ 5 = 21,565 - No remainder! 5 is one of the factors!
21,565 ÷ 5 = 4,313 - No remainder! 5 is one of the factors!
4,313 ÷ 5 = 862.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
4,313 ÷ 7 = 616.1429 - This has a remainder. 7 is not a factor.
4,313 ÷ 11 = 392.0909 - This has a remainder. 11 is not a factor.
4,313 ÷ 13 = 331.7692 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,313 ÷ 19 = 227 - No remainder! 19 is one of the factors!
227 ÷ 19 = 11.9474 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
227 ÷ 23 = 9.8696 - This has a remainder. 23 is not a factor.
227 ÷ 29 = 7.8276 - This has a remainder. 29 is not a factor.
227 ÷ 31 = 7.3226 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
227 ÷ 227 = 1 - No remainder! 227 is one of the factors!

The orange divisor(s) above are the prime factors of the number 82,809,600. 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 3 x 5 x 5 x 19 x 227 = 82,809,600. It can also be written in exponential form as 28 x 31 x 52 x 191 x 2271.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 82,809,600.

82,809,600
Factor Arrows
241,404,800
Factor Arrows
220,702,400
Factor Arrows
210,351,200
Factor Arrows
25,175,600
Factor Arrows
22,587,800
Factor Arrows
21,293,900
Factor Arrows
2646,950
Factor Arrows
2323,475
Factor Arrows
3107,825
Factor Arrows
521,565
Factor Arrows
54,313
Factor Arrows
19227

More Prime Factorization Examples

82,809,59882,809,59982,809,60182,809,602
21 x 831 x 2331 x 2,14111,9971 x 41,467171 x 171 x 695,879121 x 41,404,8011

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