Q: What is the prime factorization of the number 102,313,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 5 x 5 x 11 x 23 x 337
    • or also written as { 2, 2, 2, 2, 3, 5, 5, 11, 23, 337 }
  • Written in exponential form: 24 x 31 x 52 x 111 x 231 x 3371

Why is the prime factorization of 102,313,200 written as 24 x 31 x 52 x 111 x 231 x 3371?

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 102,313,200

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 102,313,200 by 2

102,313,200 ÷ 2 = 51,156,600 - No remainder! 2 is one of the factors!
51,156,600 ÷ 2 = 25,578,300 - No remainder! 2 is one of the factors!
25,578,300 ÷ 2 = 12,789,150 - No remainder! 2 is one of the factors!
12,789,150 ÷ 2 = 6,394,575 - No remainder! 2 is one of the factors!
6,394,575 ÷ 2 = 3,197,287.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,394,575 ÷ 3 = 2,131,525 - No remainder! 3 is one of the factors!
2,131,525 ÷ 3 = 710,508.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,131,525 ÷ 5 = 426,305 - No remainder! 5 is one of the factors!
426,305 ÷ 5 = 85,261 - No remainder! 5 is one of the factors!
85,261 ÷ 5 = 17,052.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
85,261 ÷ 7 = 12,180.1429 - This has a remainder. 7 is not a factor.
85,261 ÷ 11 = 7,751 - No remainder! 11 is one of the factors!
7,751 ÷ 11 = 704.6364 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
7,751 ÷ 13 = 596.2308 - This has a remainder. 13 is not a factor.
7,751 ÷ 17 = 455.9412 - This has a remainder. 17 is not a factor.
7,751 ÷ 19 = 407.9474 - This has a remainder. 19 is not a factor.
7,751 ÷ 23 = 337 - No remainder! 23 is one of the factors!
337 ÷ 23 = 14.6522 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
337 ÷ 29 = 11.6207 - This has a remainder. 29 is not a factor.
337 ÷ 31 = 10.871 - This has a remainder. 31 is not a factor.
337 ÷ 37 = 9.1081 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
337 ÷ 337 = 1 - No remainder! 337 is one of the factors!

The orange divisor(s) above are the prime factors of the number 102,313,200. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 5 x 5 x 11 x 23 x 337 = 102,313,200. It can also be written in exponential form as 24 x 31 x 52 x 111 x 231 x 3371.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 102,313,200.

102,313,200
Factor Arrows
251,156,600
Factor Arrows
225,578,300
Factor Arrows
212,789,150
Factor Arrows
26,394,575
Factor Arrows
32,131,525
Factor Arrows
5426,305
Factor Arrows
585,261
Factor Arrows
117,751
Factor Arrows
23337

More Prime Factorization Examples

102,313,198102,313,199102,313,201102,313,202
21 x 131 x 591 x 66,6971102,313,1991711 x 1,441,031121 x 1011 x 506,5011

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