Q: What is the prime factorization of the number 281,313,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5 x 5 x 23 x 151
    • or also written as { 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 23, 151 }
  • Written in exponential form: 23 x 34 x 53 x 231 x 1511

Why is the prime factorization of 281,313,000 written as 23 x 34 x 53 x 231 x 1511?

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 281,313,000

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 281,313,000 by 2

281,313,000 ÷ 2 = 140,656,500 - No remainder! 2 is one of the factors!
140,656,500 ÷ 2 = 70,328,250 - No remainder! 2 is one of the factors!
70,328,250 ÷ 2 = 35,164,125 - No remainder! 2 is one of the factors!
35,164,125 ÷ 2 = 17,582,062.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
35,164,125 ÷ 3 = 11,721,375 - No remainder! 3 is one of the factors!
11,721,375 ÷ 3 = 3,907,125 - No remainder! 3 is one of the factors!
3,907,125 ÷ 3 = 1,302,375 - No remainder! 3 is one of the factors!
1,302,375 ÷ 3 = 434,125 - No remainder! 3 is one of the factors!
434,125 ÷ 3 = 144,708.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
434,125 ÷ 5 = 86,825 - No remainder! 5 is one of the factors!
86,825 ÷ 5 = 17,365 - No remainder! 5 is one of the factors!
17,365 ÷ 5 = 3,473 - No remainder! 5 is one of the factors!
3,473 ÷ 5 = 694.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
3,473 ÷ 7 = 496.1429 - This has a remainder. 7 is not a factor.
3,473 ÷ 11 = 315.7273 - This has a remainder. 11 is not a factor.
3,473 ÷ 13 = 267.1538 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,473 ÷ 23 = 151 - No remainder! 23 is one of the factors!
151 ÷ 23 = 6.5652 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
151 ÷ 29 = 5.2069 - This has a remainder. 29 is not a factor.
151 ÷ 31 = 4.871 - This has a remainder. 31 is not a factor.
151 ÷ 37 = 4.0811 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
151 ÷ 151 = 1 - No remainder! 151 is one of the factors!

The orange divisor(s) above are the prime factors of the number 281,313,000. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5 x 5 x 23 x 151 = 281,313,000. It can also be written in exponential form as 23 x 34 x 53 x 231 x 1511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 281,313,000.

281,313,000
Factor Arrows
2140,656,500
Factor Arrows
270,328,250
Factor Arrows
235,164,125
Factor Arrows
311,721,375
Factor Arrows
33,907,125
Factor Arrows
31,302,375
Factor Arrows
3434,125
Factor Arrows
586,825
Factor Arrows
517,365
Factor Arrows
53,473
Factor Arrows
23151

More Prime Factorization Examples

281,312,998281,312,999281,313,001281,313,002
21 x 371 x 971 x 39,1911111 x 1791 x 142,8711471 x 1271 x 47,129121 x 140,656,5011

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