Q: What is the prime factorization of the number 110,066,096?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 7 x 607 x 1,619
    • or also written as { 2, 2, 2, 2, 7, 607, 1,619 }
  • Written in exponential form: 24 x 71 x 6071 x 1,6191

Why is the prime factorization of 110,066,096 written as 24 x 71 x 6071 x 1,6191?

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 110,066,096

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 110,066,096 by 2

110,066,096 ÷ 2 = 55,033,048 - No remainder! 2 is one of the factors!
55,033,048 ÷ 2 = 27,516,524 - No remainder! 2 is one of the factors!
27,516,524 ÷ 2 = 13,758,262 - No remainder! 2 is one of the factors!
13,758,262 ÷ 2 = 6,879,131 - No remainder! 2 is one of the factors!
6,879,131 ÷ 2 = 3,439,565.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,879,131 ÷ 3 = 2,293,043.6667 - This has a remainder. 3 is not a factor.
6,879,131 ÷ 5 = 1,375,826.2 - This has a remainder. 5 is not a factor.
6,879,131 ÷ 7 = 982,733 - No remainder! 7 is one of the factors!
982,733 ÷ 7 = 140,390.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
982,733 ÷ 11 = 89,339.3636 - This has a remainder. 11 is not a factor.
982,733 ÷ 13 = 75,594.8462 - This has a remainder. 13 is not a factor.
982,733 ÷ 17 = 57,807.8235 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
982,733 ÷ 607 = 1,619 - No remainder! 607 is one of the factors!
1,619 ÷ 607 = 2.6672 - There is a remainder. We can't divide by 607 evenly anymore. Let's try the next prime number
1,619 ÷ 613 = 2.6411 - This has a remainder. 613 is not a factor.
1,619 ÷ 617 = 2.624 - This has a remainder. 617 is not a factor.
1,619 ÷ 619 = 2.6155 - This has a remainder. 619 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,619 ÷ 1,619 = 1 - No remainder! 1,619 is one of the factors!

The orange divisor(s) above are the prime factors of the number 110,066,096. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 7 x 607 x 1,619 = 110,066,096. It can also be written in exponential form as 24 x 71 x 6071 x 1,6191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 110,066,096.

110,066,096
Factor Arrows
255,033,048
Factor Arrows
227,516,524
Factor Arrows
213,758,262
Factor Arrows
26,879,131
Factor Arrows
7982,733
Factor Arrows
6071,619

More Prime Factorization Examples

110,066,094110,066,095110,066,097110,066,098
21 x 33 x 971 x 21,013151 x 3,3471 x 6,577131 x 2,2211 x 16,519121 x 55,033,0491

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