Q: What is the prime factorization of the number 127,331,328?

 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 3 x 181 x 229
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 181, 229 }
  • Written in exponential form: 210 x 31 x 1811 x 2291

Why is the prime factorization of 127,331,328 written as 210 x 31 x 1811 x 2291?

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 127,331,328

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 127,331,328 by 2

127,331,328 ÷ 2 = 63,665,664 - No remainder! 2 is one of the factors!
63,665,664 ÷ 2 = 31,832,832 - No remainder! 2 is one of the factors!
31,832,832 ÷ 2 = 15,916,416 - No remainder! 2 is one of the factors!
15,916,416 ÷ 2 = 7,958,208 - No remainder! 2 is one of the factors!
7,958,208 ÷ 2 = 3,979,104 - No remainder! 2 is one of the factors!
3,979,104 ÷ 2 = 1,989,552 - No remainder! 2 is one of the factors!
1,989,552 ÷ 2 = 994,776 - No remainder! 2 is one of the factors!
994,776 ÷ 2 = 497,388 - No remainder! 2 is one of the factors!
497,388 ÷ 2 = 248,694 - No remainder! 2 is one of the factors!
248,694 ÷ 2 = 124,347 - No remainder! 2 is one of the factors!
124,347 ÷ 2 = 62,173.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
124,347 ÷ 3 = 41,449 - No remainder! 3 is one of the factors!
41,449 ÷ 3 = 13,816.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
41,449 ÷ 5 = 8,289.8 - This has a remainder. 5 is not a factor.
41,449 ÷ 7 = 5,921.2857 - This has a remainder. 7 is not a factor.
41,449 ÷ 11 = 3,768.0909 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
41,449 ÷ 181 = 229 - No remainder! 181 is one of the factors!
229 ÷ 181 = 1.2652 - There is a remainder. We can't divide by 181 evenly anymore. Let's try the next prime number
229 ÷ 191 = 1.199 - This has a remainder. 191 is not a factor.
229 ÷ 193 = 1.1865 - This has a remainder. 193 is not a factor.
229 ÷ 197 = 1.1624 - This has a remainder. 197 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
229 ÷ 229 = 1 - No remainder! 229 is one of the factors!

The orange divisor(s) above are the prime factors of the number 127,331,328. 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 3 x 181 x 229 = 127,331,328. It can also be written in exponential form as 210 x 31 x 1811 x 2291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 127,331,328.

127,331,328
Factor Arrows
263,665,664
Factor Arrows
231,832,832
Factor Arrows
215,916,416
Factor Arrows
27,958,208
Factor Arrows
23,979,104
Factor Arrows
21,989,552
Factor Arrows
2994,776
Factor Arrows
2497,388
Factor Arrows
2248,694
Factor Arrows
2124,347
Factor Arrows
341,449
Factor Arrows
181229

More Prime Factorization Examples

127,331,326127,331,327127,331,329127,331,330
21 x 171 x 2111 x 17,74914991 x 255,1731711 x 1,793,399121 x 51 x 71 x 1,2911 x 1,4091

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