Q: What is the prime factorization of the number 111,230,112?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 7 x 103 x 1,607
    • or also written as { 2, 2, 2, 2, 2, 3, 7, 103, 1,607 }
  • Written in exponential form: 25 x 31 x 71 x 1031 x 1,6071

Why is the prime factorization of 111,230,112 written as 25 x 31 x 71 x 1031 x 1,6071?

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 111,230,112

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 111,230,112 by 2

111,230,112 ÷ 2 = 55,615,056 - No remainder! 2 is one of the factors!
55,615,056 ÷ 2 = 27,807,528 - No remainder! 2 is one of the factors!
27,807,528 ÷ 2 = 13,903,764 - No remainder! 2 is one of the factors!
13,903,764 ÷ 2 = 6,951,882 - No remainder! 2 is one of the factors!
6,951,882 ÷ 2 = 3,475,941 - No remainder! 2 is one of the factors!
3,475,941 ÷ 2 = 1,737,970.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
3,475,941 ÷ 3 = 1,158,647 - No remainder! 3 is one of the factors!
1,158,647 ÷ 3 = 386,215.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,158,647 ÷ 5 = 231,729.4 - This has a remainder. 5 is not a factor.
1,158,647 ÷ 7 = 165,521 - No remainder! 7 is one of the factors!
165,521 ÷ 7 = 23,645.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
165,521 ÷ 11 = 15,047.3636 - This has a remainder. 11 is not a factor.
165,521 ÷ 13 = 12,732.3846 - This has a remainder. 13 is not a factor.
165,521 ÷ 17 = 9,736.5294 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
165,521 ÷ 103 = 1,607 - No remainder! 103 is one of the factors!
1,607 ÷ 103 = 15.6019 - There is a remainder. We can't divide by 103 evenly anymore. Let's try the next prime number
1,607 ÷ 107 = 15.0187 - This has a remainder. 107 is not a factor.
1,607 ÷ 109 = 14.7431 - This has a remainder. 109 is not a factor.
1,607 ÷ 113 = 14.2212 - This has a remainder. 113 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,607 ÷ 1,607 = 1 - No remainder! 1,607 is one of the factors!

The orange divisor(s) above are the prime factors of the number 111,230,112. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 7 x 103 x 1,607 = 111,230,112. It can also be written in exponential form as 25 x 31 x 71 x 1031 x 1,6071.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 111,230,112.

111,230,112
Factor Arrows
255,615,056
Factor Arrows
227,807,528
Factor Arrows
213,903,764
Factor Arrows
26,951,882
Factor Arrows
23,475,941
Factor Arrows
31,158,647
Factor Arrows
7165,521
Factor Arrows
1031,607

More Prime Factorization Examples

111,230,110111,230,111111,230,113111,230,114
21 x 51 x 11,123,01111811 x 614,5311111,230,113121 x 2571 x 216,4011

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