Q: What is the prime factorization of the number 111,147,264?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 19 x 2,539
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 19, 2,539 }
  • Written in exponential form: 28 x 32 x 191 x 2,5391

Why is the prime factorization of 111,147,264 written as 28 x 32 x 191 x 2,5391?

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 111,147,264

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 111,147,264 by 2

111,147,264 ÷ 2 = 55,573,632 - No remainder! 2 is one of the factors!
55,573,632 ÷ 2 = 27,786,816 - No remainder! 2 is one of the factors!
27,786,816 ÷ 2 = 13,893,408 - No remainder! 2 is one of the factors!
13,893,408 ÷ 2 = 6,946,704 - No remainder! 2 is one of the factors!
6,946,704 ÷ 2 = 3,473,352 - No remainder! 2 is one of the factors!
3,473,352 ÷ 2 = 1,736,676 - No remainder! 2 is one of the factors!
1,736,676 ÷ 2 = 868,338 - No remainder! 2 is one of the factors!
868,338 ÷ 2 = 434,169 - No remainder! 2 is one of the factors!
434,169 ÷ 2 = 217,084.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
434,169 ÷ 3 = 144,723 - No remainder! 3 is one of the factors!
144,723 ÷ 3 = 48,241 - No remainder! 3 is one of the factors!
48,241 ÷ 3 = 16,080.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
48,241 ÷ 5 = 9,648.2 - This has a remainder. 5 is not a factor.
48,241 ÷ 7 = 6,891.5714 - This has a remainder. 7 is not a factor.
48,241 ÷ 11 = 4,385.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
48,241 ÷ 19 = 2,539 - No remainder! 19 is one of the factors!
2,539 ÷ 19 = 133.6316 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
2,539 ÷ 23 = 110.3913 - This has a remainder. 23 is not a factor.
2,539 ÷ 29 = 87.5517 - This has a remainder. 29 is not a factor.
2,539 ÷ 31 = 81.9032 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,539 ÷ 2,539 = 1 - No remainder! 2,539 is one of the factors!

The orange divisor(s) above are the prime factors of the number 111,147,264. 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 2 x 3 x 3 x 19 x 2,539 = 111,147,264. It can also be written in exponential form as 28 x 32 x 191 x 2,5391.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 111,147,264.

111,147,264
Factor Arrows
255,573,632
Factor Arrows
227,786,816
Factor Arrows
213,893,408
Factor Arrows
26,946,704
Factor Arrows
23,473,352
Factor Arrows
21,736,676
Factor Arrows
2868,338
Factor Arrows
2434,169
Factor Arrows
3144,723
Factor Arrows
348,241
Factor Arrows
192,539

More Prime Factorization Examples

111,147,262111,147,263111,147,265111,147,266
21 x 55,573,6311111,147,263151 x 22,229,453121 x 55,573,6331

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