Q: What is the prime factorization of the number 3,244,320?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 751
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 3, 5, 751 }
  • Written in exponential form: 25 x 33 x 51 x 7511

Why is the prime factorization of 3,244,320 written as 25 x 33 x 51 x 7511?

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 3,244,320

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 3,244,320 by 2

3,244,320 ÷ 2 = 1,622,160 - No remainder! 2 is one of the factors!
1,622,160 ÷ 2 = 811,080 - No remainder! 2 is one of the factors!
811,080 ÷ 2 = 405,540 - No remainder! 2 is one of the factors!
405,540 ÷ 2 = 202,770 - No remainder! 2 is one of the factors!
202,770 ÷ 2 = 101,385 - No remainder! 2 is one of the factors!
101,385 ÷ 2 = 50,692.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
101,385 ÷ 3 = 33,795 - No remainder! 3 is one of the factors!
33,795 ÷ 3 = 11,265 - No remainder! 3 is one of the factors!
11,265 ÷ 3 = 3,755 - No remainder! 3 is one of the factors!
3,755 ÷ 3 = 1,251.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,755 ÷ 5 = 751 - No remainder! 5 is one of the factors!
751 ÷ 5 = 150.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
751 ÷ 7 = 107.2857 - This has a remainder. 7 is not a factor.
751 ÷ 11 = 68.2727 - This has a remainder. 11 is not a factor.
751 ÷ 13 = 57.7692 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
751 ÷ 751 = 1 - No remainder! 751 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,244,320. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 751 = 3,244,320. It can also be written in exponential form as 25 x 33 x 51 x 7511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,244,320.

3,244,320
Factor Arrows
21,622,160
Factor Arrows
2811,080
Factor Arrows
2405,540
Factor Arrows
2202,770
Factor Arrows
2101,385
Factor Arrows
333,795
Factor Arrows
311,265
Factor Arrows
33,755
Factor Arrows
5751

More Prime Factorization Examples

3,244,3183,244,3193,244,3213,244,322
21 x 71 x 111 x 21,0671131 x 249,56318271 x 3,923121 x 1011 x 16,0611

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