Q: What is the prime factorization of the number 1,006,656?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 7 x 107
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 7, 7, 107 }
  • Written in exponential form: 26 x 31 x 72 x 1071

Why is the prime factorization of 1,006,656 written as 26 x 31 x 72 x 1071?

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 1,006,656

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 1,006,656 by 2

1,006,656 ÷ 2 = 503,328 - No remainder! 2 is one of the factors!
503,328 ÷ 2 = 251,664 - No remainder! 2 is one of the factors!
251,664 ÷ 2 = 125,832 - No remainder! 2 is one of the factors!
125,832 ÷ 2 = 62,916 - No remainder! 2 is one of the factors!
62,916 ÷ 2 = 31,458 - No remainder! 2 is one of the factors!
31,458 ÷ 2 = 15,729 - No remainder! 2 is one of the factors!
15,729 ÷ 2 = 7,864.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
15,729 ÷ 3 = 5,243 - No remainder! 3 is one of the factors!
5,243 ÷ 3 = 1,747.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
5,243 ÷ 5 = 1,048.6 - This has a remainder. 5 is not a factor.
5,243 ÷ 7 = 749 - No remainder! 7 is one of the factors!
749 ÷ 7 = 107 - No remainder! 7 is one of the factors!
107 ÷ 7 = 15.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
107 ÷ 11 = 9.7273 - This has a remainder. 11 is not a factor.
107 ÷ 13 = 8.2308 - This has a remainder. 13 is not a factor.
107 ÷ 17 = 6.2941 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
107 ÷ 107 = 1 - No remainder! 107 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,006,656. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 7 x 107 = 1,006,656. It can also be written in exponential form as 26 x 31 x 72 x 1071.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,006,656.

1,006,656
Factor Arrows
2503,328
Factor Arrows
2251,664
Factor Arrows
2125,832
Factor Arrows
262,916
Factor Arrows
231,458
Factor Arrows
215,729
Factor Arrows
35,243
Factor Arrows
7749
Factor Arrows
7107

More Prime Factorization Examples

1,006,6541,006,6551,006,6571,006,658
21 x 111 x 45,757151 x 131 x 171 x 91112411 x 4,177121 x 191 x 591 x 4491

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