Q: What is the prime factorization of the number 110,511,600?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 5 x 5 x 19 x 37 x 131
    • or also written as { 2, 2, 2, 2, 3, 5, 5, 19, 37, 131 }
  • Written in exponential form: 24 x 31 x 52 x 191 x 371 x 1311

Why is the prime factorization of 110,511,600 written as 24 x 31 x 52 x 191 x 371 x 1311?

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 110,511,600

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 110,511,600 by 2

110,511,600 ÷ 2 = 55,255,800 - No remainder! 2 is one of the factors!
55,255,800 ÷ 2 = 27,627,900 - No remainder! 2 is one of the factors!
27,627,900 ÷ 2 = 13,813,950 - No remainder! 2 is one of the factors!
13,813,950 ÷ 2 = 6,906,975 - No remainder! 2 is one of the factors!
6,906,975 ÷ 2 = 3,453,487.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,906,975 ÷ 3 = 2,302,325 - No remainder! 3 is one of the factors!
2,302,325 ÷ 3 = 767,441.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,302,325 ÷ 5 = 460,465 - No remainder! 5 is one of the factors!
460,465 ÷ 5 = 92,093 - No remainder! 5 is one of the factors!
92,093 ÷ 5 = 18,418.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
92,093 ÷ 7 = 13,156.1429 - This has a remainder. 7 is not a factor.
92,093 ÷ 11 = 8,372.0909 - This has a remainder. 11 is not a factor.
92,093 ÷ 13 = 7,084.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
92,093 ÷ 19 = 4,847 - No remainder! 19 is one of the factors!
4,847 ÷ 19 = 255.1053 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
4,847 ÷ 23 = 210.7391 - This has a remainder. 23 is not a factor.
4,847 ÷ 29 = 167.1379 - This has a remainder. 29 is not a factor.
4,847 ÷ 31 = 156.3548 - This has a remainder. 31 is not a factor.
4,847 ÷ 37 = 131 - No remainder! 37 is one of the factors!
131 ÷ 37 = 3.5405 - There is a remainder. We can't divide by 37 evenly anymore. Let's try the next prime number
131 ÷ 41 = 3.1951 - This has a remainder. 41 is not a factor.
131 ÷ 43 = 3.0465 - This has a remainder. 43 is not a factor.
131 ÷ 47 = 2.7872 - This has a remainder. 47 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
131 ÷ 131 = 1 - No remainder! 131 is one of the factors!

The orange divisor(s) above are the prime factors of the number 110,511,600. 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 19 x 37 x 131 = 110,511,600. It can also be written in exponential form as 24 x 31 x 52 x 191 x 371 x 1311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 110,511,600.

110,511,600
Factor Arrows
255,255,800
Factor Arrows
227,627,900
Factor Arrows
213,813,950
Factor Arrows
26,906,975
Factor Arrows
32,302,325
Factor Arrows
5460,465
Factor Arrows
592,093
Factor Arrows
194,847
Factor Arrows
37131

More Prime Factorization Examples

110,511,598110,511,599110,511,601110,511,602
21 x 1371 x 403,3271113 x 791 x 1,05119071 x 121,843121 x 591 x 936,5391

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