Q: What is the prime factorization of the number 724,224?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 23 x 41
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 23, 41 }
  • Written in exponential form: 28 x 31 x 231 x 411

Why is the prime factorization of 724,224 written as 28 x 31 x 231 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 724,224

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 724,224 by 2

724,224 ÷ 2 = 362,112 - No remainder! 2 is one of the factors!
362,112 ÷ 2 = 181,056 - No remainder! 2 is one of the factors!
181,056 ÷ 2 = 90,528 - No remainder! 2 is one of the factors!
90,528 ÷ 2 = 45,264 - No remainder! 2 is one of the factors!
45,264 ÷ 2 = 22,632 - No remainder! 2 is one of the factors!
22,632 ÷ 2 = 11,316 - No remainder! 2 is one of the factors!
11,316 ÷ 2 = 5,658 - No remainder! 2 is one of the factors!
5,658 ÷ 2 = 2,829 - No remainder! 2 is one of the factors!
2,829 ÷ 2 = 1,414.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,829 ÷ 3 = 943 - No remainder! 3 is one of the factors!
943 ÷ 3 = 314.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
943 ÷ 5 = 188.6 - This has a remainder. 5 is not a factor.
943 ÷ 7 = 134.7143 - This has a remainder. 7 is not a factor.
943 ÷ 11 = 85.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
943 ÷ 23 = 41 - No remainder! 23 is one of the factors!
41 ÷ 23 = 1.7826 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
41 ÷ 29 = 1.4138 - This has a remainder. 29 is not a factor.
41 ÷ 31 = 1.3226 - This has a remainder. 31 is not a factor.
41 ÷ 37 = 1.1081 - This has a remainder. 37 is not a factor.
41 ÷ 41 = 1 - No remainder! 41 is one of the factors!

The orange divisor(s) above are the prime factors of the number 724,224. 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 23 x 41 = 724,224. It can also be written in exponential form as 28 x 31 x 231 x 411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 724,224.

724,224
Factor Arrows
2362,112
Factor Arrows
2181,056
Factor Arrows
290,528
Factor Arrows
245,264
Factor Arrows
222,632
Factor Arrows
211,316
Factor Arrows
25,658
Factor Arrows
22,829
Factor Arrows
3943
Factor Arrows
2341

More Prime Factorization Examples

724,222724,223724,225724,226
21 x 311 x 11,6811191 x 471 x 811152 x 591 x 491121 x 2571 x 1,4091

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