Q: What is the prime factorization of the number 331,313,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 5 x 5 x 97 x 8,539
    • or also written as { 2, 2, 2, 2, 5, 5, 97, 8,539 }
  • Written in exponential form: 24 x 52 x 971 x 8,5391

Why is the prime factorization of 331,313,200 written as 24 x 52 x 971 x 8,5391?

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 331,313,200

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 331,313,200 by 2

331,313,200 ÷ 2 = 165,656,600 - No remainder! 2 is one of the factors!
165,656,600 ÷ 2 = 82,828,300 - No remainder! 2 is one of the factors!
82,828,300 ÷ 2 = 41,414,150 - No remainder! 2 is one of the factors!
41,414,150 ÷ 2 = 20,707,075 - No remainder! 2 is one of the factors!
20,707,075 ÷ 2 = 10,353,537.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
20,707,075 ÷ 3 = 6,902,358.3333 - This has a remainder. 3 is not a factor.
20,707,075 ÷ 5 = 4,141,415 - No remainder! 5 is one of the factors!
4,141,415 ÷ 5 = 828,283 - No remainder! 5 is one of the factors!
828,283 ÷ 5 = 165,656.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
828,283 ÷ 7 = 118,326.1429 - This has a remainder. 7 is not a factor.
828,283 ÷ 11 = 75,298.4545 - This has a remainder. 11 is not a factor.
828,283 ÷ 13 = 63,714.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
828,283 ÷ 97 = 8,539 - No remainder! 97 is one of the factors!
8,539 ÷ 97 = 88.0309 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
8,539 ÷ 101 = 84.5446 - This has a remainder. 101 is not a factor.
8,539 ÷ 103 = 82.9029 - This has a remainder. 103 is not a factor.
8,539 ÷ 107 = 79.8037 - This has a remainder. 107 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
8,539 ÷ 8,539 = 1 - No remainder! 8,539 is one of the factors!

The orange divisor(s) above are the prime factors of the number 331,313,200. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 5 x 97 x 8,539 = 331,313,200. It can also be written in exponential form as 24 x 52 x 971 x 8,5391.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 331,313,200.

331,313,200
Factor Arrows
2165,656,600
Factor Arrows
282,828,300
Factor Arrows
241,414,150
Factor Arrows
220,707,075
Factor Arrows
54,141,415
Factor Arrows
5828,283
Factor Arrows
978,539

More Prime Factorization Examples

331,313,198331,313,199331,313,201331,313,202
21 x 2631 x 629,873131 x 71 x 471 x 1031 x 3,2591731 x 4,538,537121 x 32 x 111 x 591 x 791 x 3591

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