Q: What is the prime factorization of the number 386,250?

 A:
  • The prime factors are: 2 x 3 x 5 x 5 x 5 x 5 x 103
    • or also written as { 2, 3, 5, 5, 5, 5, 103 }
  • Written in exponential form: 21 x 31 x 54 x 1031

Why is the prime factorization of 386,250 written as 21 x 31 x 54 x 1031?

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 386,250

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 386,250 by 2

386,250 ÷ 2 = 193,125 - No remainder! 2 is one of the factors!
193,125 ÷ 2 = 96,562.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
193,125 ÷ 3 = 64,375 - No remainder! 3 is one of the factors!
64,375 ÷ 3 = 21,458.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
64,375 ÷ 5 = 12,875 - No remainder! 5 is one of the factors!
12,875 ÷ 5 = 2,575 - No remainder! 5 is one of the factors!
2,575 ÷ 5 = 515 - No remainder! 5 is one of the factors!
515 ÷ 5 = 103 - No remainder! 5 is one of the factors!
103 ÷ 5 = 20.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
103 ÷ 7 = 14.7143 - This has a remainder. 7 is not a factor.
103 ÷ 11 = 9.3636 - This has a remainder. 11 is not a factor.
103 ÷ 13 = 7.9231 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
103 ÷ 103 = 1 - No remainder! 103 is one of the factors!

The orange divisor(s) above are the prime factors of the number 386,250. If we put all of it together we have the factors 2 x 3 x 5 x 5 x 5 x 5 x 103 = 386,250. It can also be written in exponential form as 21 x 31 x 54 x 1031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 386,250.

386,250
Factor Arrows
2193,125
Factor Arrows
364,375
Factor Arrows
512,875
Factor Arrows
52,575
Factor Arrows
5515
Factor Arrows
5103

More Prime Factorization Examples

386,248386,249386,251386,252
23 x 48,2811386,2491191 x 291 x 701122 x 611 x 1,5831

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