Q: What is the prime factorization of the number 512,000?

 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 5 x 5 x 5
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5 }
  • Written in exponential form: 212 x 53

Why is the prime factorization of 512,000 written as 212 x 53?

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 512,000

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 512,000 by 2

512,000 ÷ 2 = 256,000 - No remainder! 2 is one of the factors!
256,000 ÷ 2 = 128,000 - No remainder! 2 is one of the factors!
128,000 ÷ 2 = 64,000 - No remainder! 2 is one of the factors!
64,000 ÷ 2 = 32,000 - No remainder! 2 is one of the factors!
32,000 ÷ 2 = 16,000 - No remainder! 2 is one of the factors!
16,000 ÷ 2 = 8,000 - No remainder! 2 is one of the factors!
8,000 ÷ 2 = 4,000 - No remainder! 2 is one of the factors!
4,000 ÷ 2 = 2,000 - No remainder! 2 is one of the factors!
2,000 ÷ 2 = 1,000 - No remainder! 2 is one of the factors!
1,000 ÷ 2 = 500 - No remainder! 2 is one of the factors!
500 ÷ 2 = 250 - No remainder! 2 is one of the factors!
250 ÷ 2 = 125 - No remainder! 2 is one of the factors!
125 ÷ 2 = 62.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
125 ÷ 3 = 41.6667 - This has a remainder. 3 is not a factor.
125 ÷ 5 = 25 - No remainder! 5 is one of the factors!
25 ÷ 5 = 5 - No remainder! 5 is one of the factors!
5 ÷ 5 = 1 - No remainder! 5 is one of the factors!

The orange divisor(s) above are the prime factors of the number 512,000. 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 5 x 5 x 5 = 512,000. It can also be written in exponential form as 212 x 53.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 512,000.

512,000
Factor Arrows
2256,000
Factor Arrows
2128,000
Factor Arrows
264,000
Factor Arrows
232,000
Factor Arrows
216,000
Factor Arrows
28,000
Factor Arrows
24,000
Factor Arrows
22,000
Factor Arrows
21,000
Factor Arrows
2500
Factor Arrows
2250
Factor Arrows
2125
Factor Arrows
525
Factor Arrows
55

More Prime Factorization Examples

511,998511,999512,001512,002
21 x 31 x 85,3331791 x 6,481135 x 72 x 43121 x 591 x 4,3391

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