Q: What is the prime factorization of the number 100,210,112?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 127 x 12,329
    • or also written as { 2, 2, 2, 2, 2, 2, 127, 12,329 }
  • Written in exponential form: 26 x 1271 x 12,3291

Why is the prime factorization of 100,210,112 written as 26 x 1271 x 12,3291?

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 100,210,112

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 100,210,112 by 2

100,210,112 ÷ 2 = 50,105,056 - No remainder! 2 is one of the factors!
50,105,056 ÷ 2 = 25,052,528 - No remainder! 2 is one of the factors!
25,052,528 ÷ 2 = 12,526,264 - No remainder! 2 is one of the factors!
12,526,264 ÷ 2 = 6,263,132 - No remainder! 2 is one of the factors!
6,263,132 ÷ 2 = 3,131,566 - No remainder! 2 is one of the factors!
3,131,566 ÷ 2 = 1,565,783 - No remainder! 2 is one of the factors!
1,565,783 ÷ 2 = 782,891.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,565,783 ÷ 3 = 521,927.6667 - This has a remainder. 3 is not a factor.
1,565,783 ÷ 5 = 313,156.6 - This has a remainder. 5 is not a factor.
1,565,783 ÷ 7 = 223,683.2857 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,565,783 ÷ 127 = 12,329 - No remainder! 127 is one of the factors!
12,329 ÷ 127 = 97.0787 - There is a remainder. We can't divide by 127 evenly anymore. Let's try the next prime number
12,329 ÷ 131 = 94.1145 - This has a remainder. 131 is not a factor.
12,329 ÷ 137 = 89.9927 - This has a remainder. 137 is not a factor.
12,329 ÷ 139 = 88.6978 - This has a remainder. 139 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
12,329 ÷ 12,329 = 1 - No remainder! 12,329 is one of the factors!

The orange divisor(s) above are the prime factors of the number 100,210,112. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 127 x 12,329 = 100,210,112. It can also be written in exponential form as 26 x 1271 x 12,3291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 100,210,112.

100,210,112
Factor Arrows
250,105,056
Factor Arrows
225,052,528
Factor Arrows
212,526,264
Factor Arrows
26,263,132
Factor Arrows
23,131,566
Factor Arrows
21,565,783
Factor Arrows
12712,329

More Prime Factorization Examples

100,210,110100,210,111100,210,113100,210,114
21 x 31 x 51 x 71 x 111 x 131 x 471 x 7111,2971 x 77,263132 x 9711 x 11,467121 x 50,105,0571

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