Q: What is the prime factorization of the number 111,213,312?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 137 x 151
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 137, 151 }
  • Written in exponential form: 28 x 31 x 71 x 1371 x 1511

Why is the prime factorization of 111,213,312 written as 28 x 31 x 71 x 1371 x 1511?

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 111,213,312

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 111,213,312 by 2

111,213,312 ÷ 2 = 55,606,656 - No remainder! 2 is one of the factors!
55,606,656 ÷ 2 = 27,803,328 - No remainder! 2 is one of the factors!
27,803,328 ÷ 2 = 13,901,664 - No remainder! 2 is one of the factors!
13,901,664 ÷ 2 = 6,950,832 - No remainder! 2 is one of the factors!
6,950,832 ÷ 2 = 3,475,416 - No remainder! 2 is one of the factors!
3,475,416 ÷ 2 = 1,737,708 - No remainder! 2 is one of the factors!
1,737,708 ÷ 2 = 868,854 - No remainder! 2 is one of the factors!
868,854 ÷ 2 = 434,427 - No remainder! 2 is one of the factors!
434,427 ÷ 2 = 217,213.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
434,427 ÷ 3 = 144,809 - No remainder! 3 is one of the factors!
144,809 ÷ 3 = 48,269.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
144,809 ÷ 5 = 28,961.8 - This has a remainder. 5 is not a factor.
144,809 ÷ 7 = 20,687 - No remainder! 7 is one of the factors!
20,687 ÷ 7 = 2,955.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
20,687 ÷ 11 = 1,880.6364 - This has a remainder. 11 is not a factor.
20,687 ÷ 13 = 1,591.3077 - This has a remainder. 13 is not a factor.
20,687 ÷ 17 = 1,216.8824 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
20,687 ÷ 137 = 151 - No remainder! 137 is one of the factors!
151 ÷ 137 = 1.1022 - There is a remainder. We can't divide by 137 evenly anymore. Let's try the next prime number
151 ÷ 139 = 1.0863 - This has a remainder. 139 is not a factor.
151 ÷ 149 = 1.0134 - This has a remainder. 149 is not a factor.
151 ÷ 151 = 1 - No remainder! 151 is one of the factors!

The orange divisor(s) above are the prime factors of the number 111,213,312. 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 7 x 137 x 151 = 111,213,312. It can also be written in exponential form as 28 x 31 x 71 x 1371 x 1511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 111,213,312.

111,213,312
Factor Arrows
255,606,656
Factor Arrows
227,803,328
Factor Arrows
213,901,664
Factor Arrows
26,950,832
Factor Arrows
23,475,416
Factor Arrows
21,737,708
Factor Arrows
2868,854
Factor Arrows
2434,427
Factor Arrows
3144,809
Factor Arrows
720,687
Factor Arrows
137151

More Prime Factorization Examples

111,213,310111,213,311111,213,313111,213,314
21 x 51 x 131 x 731 x 11,7191111 x 1,3811 x 7,3211971 x 1,146,529121 x 55,606,6571

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