Q: What is the prime factorization of the number 503,808?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 41
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 41 }
  • Written in exponential form: 212 x 31 x 411

Why is the prime factorization of 503,808 written as 212 x 31 x 411?

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 503,808

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 503,808 by 2

503,808 ÷ 2 = 251,904 - No remainder! 2 is one of the factors!
251,904 ÷ 2 = 125,952 - No remainder! 2 is one of the factors!
125,952 ÷ 2 = 62,976 - No remainder! 2 is one of the factors!
62,976 ÷ 2 = 31,488 - No remainder! 2 is one of the factors!
31,488 ÷ 2 = 15,744 - No remainder! 2 is one of the factors!
15,744 ÷ 2 = 7,872 - No remainder! 2 is one of the factors!
7,872 ÷ 2 = 3,936 - No remainder! 2 is one of the factors!
3,936 ÷ 2 = 1,968 - No remainder! 2 is one of the factors!
1,968 ÷ 2 = 984 - No remainder! 2 is one of the factors!
984 ÷ 2 = 492 - No remainder! 2 is one of the factors!
492 ÷ 2 = 246 - No remainder! 2 is one of the factors!
246 ÷ 2 = 123 - No remainder! 2 is one of the factors!
123 ÷ 2 = 61.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
123 ÷ 3 = 41 - No remainder! 3 is one of the factors!
41 ÷ 3 = 13.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
41 ÷ 5 = 8.2 - This has a remainder. 5 is not a factor.
41 ÷ 7 = 5.8571 - This has a remainder. 7 is not a factor.
41 ÷ 11 = 3.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
41 ÷ 41 = 1 - No remainder! 41 is one of the factors!

The orange divisor(s) above are the prime factors of the number 503,808. 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 2 x 2 x 2 x 2 x 3 x 41 = 503,808. It can also be written in exponential form as 212 x 31 x 411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 503,808.

503,808
Factor Arrows
2251,904
Factor Arrows
2125,952
Factor Arrows
262,976
Factor Arrows
231,488
Factor Arrows
215,744
Factor Arrows
27,872
Factor Arrows
23,936
Factor Arrows
21,968
Factor Arrows
2984
Factor Arrows
2492
Factor Arrows
2246
Factor Arrows
2123
Factor Arrows
341

More Prime Factorization Examples

503,806503,807503,809503,810
21 x 251,90315211 x 96711271 x 3,967121 x 51 x 831 x 6071

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