Q: What is the prime factorization of the number 426,630,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 7 x 89 x 107
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 7, 89, 107 }
  • Written in exponential form: 28 x 52 x 71 x 891 x 1071

Why is the prime factorization of 426,630,400 written as 28 x 52 x 71 x 891 x 1071?

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 426,630,400

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 426,630,400 by 2

426,630,400 ÷ 2 = 213,315,200 - No remainder! 2 is one of the factors!
213,315,200 ÷ 2 = 106,657,600 - No remainder! 2 is one of the factors!
106,657,600 ÷ 2 = 53,328,800 - No remainder! 2 is one of the factors!
53,328,800 ÷ 2 = 26,664,400 - No remainder! 2 is one of the factors!
26,664,400 ÷ 2 = 13,332,200 - No remainder! 2 is one of the factors!
13,332,200 ÷ 2 = 6,666,100 - No remainder! 2 is one of the factors!
6,666,100 ÷ 2 = 3,333,050 - No remainder! 2 is one of the factors!
3,333,050 ÷ 2 = 1,666,525 - No remainder! 2 is one of the factors!
1,666,525 ÷ 2 = 833,262.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,666,525 ÷ 3 = 555,508.3333 - This has a remainder. 3 is not a factor.
1,666,525 ÷ 5 = 333,305 - No remainder! 5 is one of the factors!
333,305 ÷ 5 = 66,661 - No remainder! 5 is one of the factors!
66,661 ÷ 5 = 13,332.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
66,661 ÷ 7 = 9,523 - No remainder! 7 is one of the factors!
9,523 ÷ 7 = 1,360.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
9,523 ÷ 11 = 865.7273 - This has a remainder. 11 is not a factor.
9,523 ÷ 13 = 732.5385 - This has a remainder. 13 is not a factor.
9,523 ÷ 17 = 560.1765 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
9,523 ÷ 89 = 107 - No remainder! 89 is one of the factors!
107 ÷ 89 = 1.2022 - There is a remainder. We can't divide by 89 evenly anymore. Let's try the next prime number
107 ÷ 97 = 1.1031 - This has a remainder. 97 is not a factor.
107 ÷ 101 = 1.0594 - This has a remainder. 101 is not a factor.
107 ÷ 103 = 1.0388 - This has a remainder. 103 is not a factor.
107 ÷ 107 = 1 - No remainder! 107 is one of the factors!

The orange divisor(s) above are the prime factors of the number 426,630,400. 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 5 x 5 x 7 x 89 x 107 = 426,630,400. It can also be written in exponential form as 28 x 52 x 71 x 891 x 1071.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 426,630,400.

426,630,400
Factor Arrows
2213,315,200
Factor Arrows
2106,657,600
Factor Arrows
253,328,800
Factor Arrows
226,664,400
Factor Arrows
213,332,200
Factor Arrows
26,666,100
Factor Arrows
23,333,050
Factor Arrows
21,666,525
Factor Arrows
5333,305
Factor Arrows
566,661
Factor Arrows
79,523
Factor Arrows
89107

More Prime Factorization Examples

426,630,398426,630,399426,630,401426,630,402
21 x 1,0491 x 203,351131 x 131 x 10,939,2411311 x 611 x 225,611121 x 34 x 111 x 171 x 14,0831

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