Q: What is the prime factorization of the number 117,705,600?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 61 x 67
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 5, 61, 67 }
  • Written in exponential form: 27 x 32 x 52 x 611 x 671

Why is the prime factorization of 117,705,600 written as 27 x 32 x 52 x 611 x 671?

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 117,705,600

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 117,705,600 by 2

117,705,600 ÷ 2 = 58,852,800 - No remainder! 2 is one of the factors!
58,852,800 ÷ 2 = 29,426,400 - No remainder! 2 is one of the factors!
29,426,400 ÷ 2 = 14,713,200 - No remainder! 2 is one of the factors!
14,713,200 ÷ 2 = 7,356,600 - No remainder! 2 is one of the factors!
7,356,600 ÷ 2 = 3,678,300 - No remainder! 2 is one of the factors!
3,678,300 ÷ 2 = 1,839,150 - No remainder! 2 is one of the factors!
1,839,150 ÷ 2 = 919,575 - No remainder! 2 is one of the factors!
919,575 ÷ 2 = 459,787.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
919,575 ÷ 3 = 306,525 - No remainder! 3 is one of the factors!
306,525 ÷ 3 = 102,175 - No remainder! 3 is one of the factors!
102,175 ÷ 3 = 34,058.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
102,175 ÷ 5 = 20,435 - No remainder! 5 is one of the factors!
20,435 ÷ 5 = 4,087 - No remainder! 5 is one of the factors!
4,087 ÷ 5 = 817.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
4,087 ÷ 7 = 583.8571 - This has a remainder. 7 is not a factor.
4,087 ÷ 11 = 371.5455 - This has a remainder. 11 is not a factor.
4,087 ÷ 13 = 314.3846 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,087 ÷ 61 = 67 - No remainder! 61 is one of the factors!
67 ÷ 61 = 1.0984 - There is a remainder. We can't divide by 61 evenly anymore. Let's try the next prime number
67 ÷ 67 = 1 - No remainder! 67 is one of the factors!

The orange divisor(s) above are the prime factors of the number 117,705,600. 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 3 x 3 x 5 x 5 x 61 x 67 = 117,705,600. It can also be written in exponential form as 27 x 32 x 52 x 611 x 671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 117,705,600.

117,705,600
Factor Arrows
258,852,800
Factor Arrows
229,426,400
Factor Arrows
214,713,200
Factor Arrows
27,356,600
Factor Arrows
23,678,300
Factor Arrows
21,839,150
Factor Arrows
2919,575
Factor Arrows
3306,525
Factor Arrows
3102,175
Factor Arrows
520,435
Factor Arrows
54,087
Factor Arrows
6167

More Prime Factorization Examples

117,705,598117,705,599117,705,601117,705,602
21 x 2711 x 217,1691111 x 10,700,5091131 x 9,054,277121 x 71 x 1011 x 83,2431

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