Q: What is the prime factorization of the number 45,552,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 13 x 73
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 5, 13, 73 }
  • Written in exponential form: 27 x 31 x 53 x 131 x 731

Why is the prime factorization of 45,552,000 written as 27 x 31 x 53 x 131 x 731?

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 45,552,000

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 45,552,000 by 2

45,552,000 ÷ 2 = 22,776,000 - No remainder! 2 is one of the factors!
22,776,000 ÷ 2 = 11,388,000 - No remainder! 2 is one of the factors!
11,388,000 ÷ 2 = 5,694,000 - No remainder! 2 is one of the factors!
5,694,000 ÷ 2 = 2,847,000 - No remainder! 2 is one of the factors!
2,847,000 ÷ 2 = 1,423,500 - No remainder! 2 is one of the factors!
1,423,500 ÷ 2 = 711,750 - No remainder! 2 is one of the factors!
711,750 ÷ 2 = 355,875 - No remainder! 2 is one of the factors!
355,875 ÷ 2 = 177,937.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
355,875 ÷ 3 = 118,625 - No remainder! 3 is one of the factors!
118,625 ÷ 3 = 39,541.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
118,625 ÷ 5 = 23,725 - No remainder! 5 is one of the factors!
23,725 ÷ 5 = 4,745 - No remainder! 5 is one of the factors!
4,745 ÷ 5 = 949 - No remainder! 5 is one of the factors!
949 ÷ 5 = 189.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
949 ÷ 7 = 135.5714 - This has a remainder. 7 is not a factor.
949 ÷ 11 = 86.2727 - This has a remainder. 11 is not a factor.
949 ÷ 13 = 73 - No remainder! 13 is one of the factors!
73 ÷ 13 = 5.6154 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
73 ÷ 17 = 4.2941 - This has a remainder. 17 is not a factor.
73 ÷ 19 = 3.8421 - This has a remainder. 19 is not a factor.
73 ÷ 23 = 3.1739 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
73 ÷ 73 = 1 - No remainder! 73 is one of the factors!

The orange divisor(s) above are the prime factors of the number 45,552,000. 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 3 x 5 x 5 x 5 x 13 x 73 = 45,552,000. It can also be written in exponential form as 27 x 31 x 53 x 131 x 731.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 45,552,000.

45,552,000
Factor Arrows
222,776,000
Factor Arrows
211,388,000
Factor Arrows
25,694,000
Factor Arrows
22,847,000
Factor Arrows
21,423,500
Factor Arrows
2711,750
Factor Arrows
2355,875
Factor Arrows
3118,625
Factor Arrows
523,725
Factor Arrows
54,745
Factor Arrows
5949
Factor Arrows
1373

More Prime Factorization Examples

45,551,99845,551,99945,552,00145,552,002
21 x 22,775,999145,551,9991111 x 4,141,091121 x 6831 x 33,3471

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