Q: What is the prime factorization of the number 26,111,232?

 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 7 x 1,619
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 7, 1,619 }
  • Written in exponential form: 28 x 32 x 71 x 1,6191

Why is the prime factorization of 26,111,232 written as 28 x 32 x 71 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 26,111,232

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 26,111,232 by 2

26,111,232 ÷ 2 = 13,055,616 - No remainder! 2 is one of the factors!
13,055,616 ÷ 2 = 6,527,808 - No remainder! 2 is one of the factors!
6,527,808 ÷ 2 = 3,263,904 - No remainder! 2 is one of the factors!
3,263,904 ÷ 2 = 1,631,952 - No remainder! 2 is one of the factors!
1,631,952 ÷ 2 = 815,976 - No remainder! 2 is one of the factors!
815,976 ÷ 2 = 407,988 - No remainder! 2 is one of the factors!
407,988 ÷ 2 = 203,994 - No remainder! 2 is one of the factors!
203,994 ÷ 2 = 101,997 - No remainder! 2 is one of the factors!
101,997 ÷ 2 = 50,998.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
101,997 ÷ 3 = 33,999 - No remainder! 3 is one of the factors!
33,999 ÷ 3 = 11,333 - No remainder! 3 is one of the factors!
11,333 ÷ 3 = 3,777.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
11,333 ÷ 5 = 2,266.6 - This has a remainder. 5 is not a factor.
11,333 ÷ 7 = 1,619 - No remainder! 7 is one of the factors!
1,619 ÷ 7 = 231.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,619 ÷ 11 = 147.1818 - This has a remainder. 11 is not a factor.
1,619 ÷ 13 = 124.5385 - This has a remainder. 13 is not a factor.
1,619 ÷ 17 = 95.2353 - This has a remainder. 17 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 26,111,232. 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 7 x 1,619 = 26,111,232. It can also be written in exponential form as 28 x 32 x 71 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 26,111,232.

26,111,232
Factor Arrows
213,055,616
Factor Arrows
26,527,808
Factor Arrows
23,263,904
Factor Arrows
21,631,952
Factor Arrows
2815,976
Factor Arrows
2407,988
Factor Arrows
2203,994
Factor Arrows
2101,997
Factor Arrows
333,999
Factor Arrows
311,333
Factor Arrows
71,619

More Prime Factorization Examples

26,111,23026,111,23126,111,23326,111,234
21 x 51 x 2,611,123126,111,2311231 x 371 x 611 x 503121 x 431 x 303,6191

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