Q: What is the prime factorization of the number 185,588?

 A:
  • The prime factors are: 2 x 2 x 13 x 43 x 83
    • or also written as { 2, 2, 13, 43, 83 }
  • Written in exponential form: 22 x 131 x 431 x 831

Why is the prime factorization of 185,588 written as 22 x 131 x 431 x 831?

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 185,588

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 185,588 by 2

185,588 ÷ 2 = 92,794 - No remainder! 2 is one of the factors!
92,794 ÷ 2 = 46,397 - No remainder! 2 is one of the factors!
46,397 ÷ 2 = 23,198.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
46,397 ÷ 3 = 15,465.6667 - This has a remainder. 3 is not a factor.
46,397 ÷ 5 = 9,279.4 - This has a remainder. 5 is not a factor.
46,397 ÷ 7 = 6,628.1429 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
46,397 ÷ 13 = 3,569 - No remainder! 13 is one of the factors!
3,569 ÷ 13 = 274.5385 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
3,569 ÷ 17 = 209.9412 - This has a remainder. 17 is not a factor.
3,569 ÷ 19 = 187.8421 - This has a remainder. 19 is not a factor.
3,569 ÷ 23 = 155.1739 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,569 ÷ 43 = 83 - No remainder! 43 is one of the factors!
83 ÷ 43 = 1.9302 - There is a remainder. We can't divide by 43 evenly anymore. Let's try the next prime number
83 ÷ 47 = 1.766 - This has a remainder. 47 is not a factor.
83 ÷ 53 = 1.566 - This has a remainder. 53 is not a factor.
83 ÷ 59 = 1.4068 - This has a remainder. 59 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
83 ÷ 83 = 1 - No remainder! 83 is one of the factors!

The orange divisor(s) above are the prime factors of the number 185,588. If we put all of it together we have the factors 2 x 2 x 13 x 43 x 83 = 185,588. It can also be written in exponential form as 22 x 131 x 431 x 831.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 185,588.

185,588
Factor Arrows
292,794
Factor Arrows
246,397
Factor Arrows
133,569
Factor Arrows
4383

More Prime Factorization Examples

185,586185,587185,589185,590
21 x 31 x 30,9311231 x 8,069132 x 171 x 1,213121 x 51 x 671 x 2771

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