Q: What is the prime factorization of the number 180,304?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 59 x 191
    • or also written as { 2, 2, 2, 2, 59, 191 }
  • Written in exponential form: 24 x 591 x 1911

Why is the prime factorization of 180,304 written as 24 x 591 x 1911?

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 180,304

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 180,304 by 2

180,304 ÷ 2 = 90,152 - No remainder! 2 is one of the factors!
90,152 ÷ 2 = 45,076 - No remainder! 2 is one of the factors!
45,076 ÷ 2 = 22,538 - No remainder! 2 is one of the factors!
22,538 ÷ 2 = 11,269 - No remainder! 2 is one of the factors!
11,269 ÷ 2 = 5,634.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
11,269 ÷ 3 = 3,756.3333 - This has a remainder. 3 is not a factor.
11,269 ÷ 5 = 2,253.8 - This has a remainder. 5 is not a factor.
11,269 ÷ 7 = 1,609.8571 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,269 ÷ 59 = 191 - No remainder! 59 is one of the factors!
191 ÷ 59 = 3.2373 - There is a remainder. We can't divide by 59 evenly anymore. Let's try the next prime number
191 ÷ 61 = 3.1311 - This has a remainder. 61 is not a factor.
191 ÷ 67 = 2.8507 - This has a remainder. 67 is not a factor.
191 ÷ 71 = 2.6901 - This has a remainder. 71 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
191 ÷ 191 = 1 - No remainder! 191 is one of the factors!

The orange divisor(s) above are the prime factors of the number 180,304. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 59 x 191 = 180,304. It can also be written in exponential form as 24 x 591 x 1911.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 180,304.

180,304
Factor Arrows
290,152
Factor Arrows
245,076
Factor Arrows
222,538
Factor Arrows
211,269
Factor Arrows
59191

More Prime Factorization Examples

180,302180,303180,305180,306
21 x 171 x 5,303131 x 60,101151 x 36,061121 x 35 x 71 x 531

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