Q: What is the prime factorization of the number 33,568,128?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 11 x 883
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 11, 883 }
  • Written in exponential form: 27 x 33 x 111 x 8831

Why is the prime factorization of 33,568,128 written as 27 x 33 x 111 x 8831?

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 33,568,128

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 33,568,128 by 2

33,568,128 ÷ 2 = 16,784,064 - No remainder! 2 is one of the factors!
16,784,064 ÷ 2 = 8,392,032 - No remainder! 2 is one of the factors!
8,392,032 ÷ 2 = 4,196,016 - No remainder! 2 is one of the factors!
4,196,016 ÷ 2 = 2,098,008 - No remainder! 2 is one of the factors!
2,098,008 ÷ 2 = 1,049,004 - No remainder! 2 is one of the factors!
1,049,004 ÷ 2 = 524,502 - No remainder! 2 is one of the factors!
524,502 ÷ 2 = 262,251 - No remainder! 2 is one of the factors!
262,251 ÷ 2 = 131,125.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
262,251 ÷ 3 = 87,417 - No remainder! 3 is one of the factors!
87,417 ÷ 3 = 29,139 - No remainder! 3 is one of the factors!
29,139 ÷ 3 = 9,713 - No remainder! 3 is one of the factors!
9,713 ÷ 3 = 3,237.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
9,713 ÷ 5 = 1,942.6 - This has a remainder. 5 is not a factor.
9,713 ÷ 7 = 1,387.5714 - This has a remainder. 7 is not a factor.
9,713 ÷ 11 = 883 - No remainder! 11 is one of the factors!
883 ÷ 11 = 80.2727 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
883 ÷ 13 = 67.9231 - This has a remainder. 13 is not a factor.
883 ÷ 17 = 51.9412 - This has a remainder. 17 is not a factor.
883 ÷ 19 = 46.4737 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
883 ÷ 883 = 1 - No remainder! 883 is one of the factors!

The orange divisor(s) above are the prime factors of the number 33,568,128. 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 3 x 3 x 3 x 11 x 883 = 33,568,128. It can also be written in exponential form as 27 x 33 x 111 x 8831.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 33,568,128.

33,568,128
Factor Arrows
216,784,064
Factor Arrows
28,392,032
Factor Arrows
24,196,016
Factor Arrows
22,098,008
Factor Arrows
21,049,004
Factor Arrows
2524,502
Factor Arrows
2262,251
Factor Arrows
387,417
Factor Arrows
329,139
Factor Arrows
39,713
Factor Arrows
11883

More Prime Factorization Examples

33,568,12633,568,12733,568,12933,568,130
21 x 16,784,0631791 x 424,913171 x 5771 x 8,311121 x 51 x 891 x 37,7171

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