Q: What is the prime factorization of the number 554,086,400?

 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 2 x 2 x 5 x 5 x 7 x 773
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 7, 773 }
  • Written in exponential form: 212 x 52 x 71 x 7731

Why is the prime factorization of 554,086,400 written as 212 x 52 x 71 x 7731?

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 554,086,400

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 554,086,400 by 2

554,086,400 ÷ 2 = 277,043,200 - No remainder! 2 is one of the factors!
277,043,200 ÷ 2 = 138,521,600 - No remainder! 2 is one of the factors!
138,521,600 ÷ 2 = 69,260,800 - No remainder! 2 is one of the factors!
69,260,800 ÷ 2 = 34,630,400 - No remainder! 2 is one of the factors!
34,630,400 ÷ 2 = 17,315,200 - No remainder! 2 is one of the factors!
17,315,200 ÷ 2 = 8,657,600 - No remainder! 2 is one of the factors!
8,657,600 ÷ 2 = 4,328,800 - No remainder! 2 is one of the factors!
4,328,800 ÷ 2 = 2,164,400 - No remainder! 2 is one of the factors!
2,164,400 ÷ 2 = 1,082,200 - No remainder! 2 is one of the factors!
1,082,200 ÷ 2 = 541,100 - No remainder! 2 is one of the factors!
541,100 ÷ 2 = 270,550 - No remainder! 2 is one of the factors!
270,550 ÷ 2 = 135,275 - No remainder! 2 is one of the factors!
135,275 ÷ 2 = 67,637.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
135,275 ÷ 3 = 45,091.6667 - This has a remainder. 3 is not a factor.
135,275 ÷ 5 = 27,055 - No remainder! 5 is one of the factors!
27,055 ÷ 5 = 5,411 - No remainder! 5 is one of the factors!
5,411 ÷ 5 = 1,082.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
5,411 ÷ 7 = 773 - No remainder! 7 is one of the factors!
773 ÷ 7 = 110.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
773 ÷ 11 = 70.2727 - This has a remainder. 11 is not a factor.
773 ÷ 13 = 59.4615 - This has a remainder. 13 is not a factor.
773 ÷ 17 = 45.4706 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
773 ÷ 773 = 1 - No remainder! 773 is one of the factors!

The orange divisor(s) above are the prime factors of the number 554,086,400. 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 2 x 2 x 5 x 5 x 7 x 773 = 554,086,400. It can also be written in exponential form as 212 x 52 x 71 x 7731.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 554,086,400.

554,086,400
Factor Arrows
2277,043,200
Factor Arrows
2138,521,600
Factor Arrows
269,260,800
Factor Arrows
234,630,400
Factor Arrows
217,315,200
Factor Arrows
28,657,600
Factor Arrows
24,328,800
Factor Arrows
22,164,400
Factor Arrows
21,082,200
Factor Arrows
2541,100
Factor Arrows
2270,550
Factor Arrows
2135,275
Factor Arrows
527,055
Factor Arrows
55,411
Factor Arrows
7773

More Prime Factorization Examples

554,086,398554,086,399554,086,401554,086,402
21 x 31 x 191 x 4,860,4071231 x 24,090,713131 x 111 x 16,790,497121 x 1931 x 1,435,4571

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