Q: What is the prime factorization of the number 605,664?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 701
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 3, 701 }
  • Written in exponential form: 25 x 33 x 7011

Why is the prime factorization of 605,664 written as 25 x 33 x 7011?

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 605,664

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 605,664 by 2

605,664 ÷ 2 = 302,832 - No remainder! 2 is one of the factors!
302,832 ÷ 2 = 151,416 - No remainder! 2 is one of the factors!
151,416 ÷ 2 = 75,708 - No remainder! 2 is one of the factors!
75,708 ÷ 2 = 37,854 - No remainder! 2 is one of the factors!
37,854 ÷ 2 = 18,927 - No remainder! 2 is one of the factors!
18,927 ÷ 2 = 9,463.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
18,927 ÷ 3 = 6,309 - No remainder! 3 is one of the factors!
6,309 ÷ 3 = 2,103 - No remainder! 3 is one of the factors!
2,103 ÷ 3 = 701 - No remainder! 3 is one of the factors!
701 ÷ 3 = 233.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
701 ÷ 5 = 140.2 - This has a remainder. 5 is not a factor.
701 ÷ 7 = 100.1429 - This has a remainder. 7 is not a factor.
701 ÷ 11 = 63.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
701 ÷ 701 = 1 - No remainder! 701 is one of the factors!

The orange divisor(s) above are the prime factors of the number 605,664. 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 701 = 605,664. It can also be written in exponential form as 25 x 33 x 7011.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 605,664.

605,664
Factor Arrows
2302,832
Factor Arrows
2151,416
Factor Arrows
275,708
Factor Arrows
237,854
Factor Arrows
218,927
Factor Arrows
36,309
Factor Arrows
32,103
Factor Arrows
3701

More Prime Factorization Examples

605,662605,663605,665605,666
21 x 302,8311191 x 1271 x 251151 x 291 x 4,177121 x 302,8331

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