Q: What is the prime factorization of the number 512,064?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 127
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 7, 127 }
  • Written in exponential form: 26 x 32 x 71 x 1271

Why is the prime factorization of 512,064 written as 26 x 32 x 71 x 1271?

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 512,064

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 512,064 by 2

512,064 ÷ 2 = 256,032 - No remainder! 2 is one of the factors!
256,032 ÷ 2 = 128,016 - No remainder! 2 is one of the factors!
128,016 ÷ 2 = 64,008 - No remainder! 2 is one of the factors!
64,008 ÷ 2 = 32,004 - No remainder! 2 is one of the factors!
32,004 ÷ 2 = 16,002 - No remainder! 2 is one of the factors!
16,002 ÷ 2 = 8,001 - No remainder! 2 is one of the factors!
8,001 ÷ 2 = 4,000.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
8,001 ÷ 3 = 2,667 - No remainder! 3 is one of the factors!
2,667 ÷ 3 = 889 - No remainder! 3 is one of the factors!
889 ÷ 3 = 296.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
889 ÷ 5 = 177.8 - This has a remainder. 5 is not a factor.
889 ÷ 7 = 127 - No remainder! 7 is one of the factors!
127 ÷ 7 = 18.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
127 ÷ 11 = 11.5455 - This has a remainder. 11 is not a factor.
127 ÷ 13 = 9.7692 - This has a remainder. 13 is not a factor.
127 ÷ 17 = 7.4706 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
127 ÷ 127 = 1 - No remainder! 127 is one of the factors!

The orange divisor(s) above are the prime factors of the number 512,064. 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 3 x 7 x 127 = 512,064. It can also be written in exponential form as 26 x 32 x 71 x 1271.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 512,064.

512,064
Factor Arrows
2256,032
Factor Arrows
2128,016
Factor Arrows
264,008
Factor Arrows
232,004
Factor Arrows
216,002
Factor Arrows
28,001
Factor Arrows
32,667
Factor Arrows
3889
Factor Arrows
7127

More Prime Factorization Examples

512,062512,063512,065512,066
21 x 256,0311971 x 5,279151 x 471 x 2,179121 x 256,0331

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