Q: What is the prime factorization of the number 54,024,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 2,251
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 5, 5, 5, 2,251 }
  • Written in exponential form: 26 x 31 x 53 x 2,2511

Why is the prime factorization of 54,024,000 written as 26 x 31 x 53 x 2,2511?

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 54,024,000

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 54,024,000 by 2

54,024,000 ÷ 2 = 27,012,000 - No remainder! 2 is one of the factors!
27,012,000 ÷ 2 = 13,506,000 - No remainder! 2 is one of the factors!
13,506,000 ÷ 2 = 6,753,000 - No remainder! 2 is one of the factors!
6,753,000 ÷ 2 = 3,376,500 - No remainder! 2 is one of the factors!
3,376,500 ÷ 2 = 1,688,250 - No remainder! 2 is one of the factors!
1,688,250 ÷ 2 = 844,125 - No remainder! 2 is one of the factors!
844,125 ÷ 2 = 422,062.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
844,125 ÷ 3 = 281,375 - No remainder! 3 is one of the factors!
281,375 ÷ 3 = 93,791.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
281,375 ÷ 5 = 56,275 - No remainder! 5 is one of the factors!
56,275 ÷ 5 = 11,255 - No remainder! 5 is one of the factors!
11,255 ÷ 5 = 2,251 - No remainder! 5 is one of the factors!
2,251 ÷ 5 = 450.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,251 ÷ 7 = 321.5714 - This has a remainder. 7 is not a factor.
2,251 ÷ 11 = 204.6364 - This has a remainder. 11 is not a factor.
2,251 ÷ 13 = 173.1538 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,251 ÷ 2,251 = 1 - No remainder! 2,251 is one of the factors!

The orange divisor(s) above are the prime factors of the number 54,024,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 2,251 = 54,024,000. It can also be written in exponential form as 26 x 31 x 53 x 2,2511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 54,024,000.

54,024,000
Factor Arrows
227,012,000
Factor Arrows
213,506,000
Factor Arrows
26,753,000
Factor Arrows
23,376,500
Factor Arrows
21,688,250
Factor Arrows
2844,125
Factor Arrows
3281,375
Factor Arrows
556,275
Factor Arrows
511,255
Factor Arrows
52,251

More Prime Factorization Examples

54,023,99854,023,99954,024,00154,024,002
21 x 71 x 1,0611 x 3,6371591 x 1671 x 5,483154,024,001121 x 3,0011 x 9,0011

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