Q: What is the prime factorization of the number 214,022,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 71 x 157
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 71, 157 }
  • Written in exponential form: 28 x 31 x 52 x 711 x 1571

Why is the prime factorization of 214,022,400 written as 28 x 31 x 52 x 711 x 1571?

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 214,022,400

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 214,022,400 by 2

214,022,400 ÷ 2 = 107,011,200 - No remainder! 2 is one of the factors!
107,011,200 ÷ 2 = 53,505,600 - No remainder! 2 is one of the factors!
53,505,600 ÷ 2 = 26,752,800 - No remainder! 2 is one of the factors!
26,752,800 ÷ 2 = 13,376,400 - No remainder! 2 is one of the factors!
13,376,400 ÷ 2 = 6,688,200 - No remainder! 2 is one of the factors!
6,688,200 ÷ 2 = 3,344,100 - No remainder! 2 is one of the factors!
3,344,100 ÷ 2 = 1,672,050 - No remainder! 2 is one of the factors!
1,672,050 ÷ 2 = 836,025 - No remainder! 2 is one of the factors!
836,025 ÷ 2 = 418,012.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
836,025 ÷ 3 = 278,675 - No remainder! 3 is one of the factors!
278,675 ÷ 3 = 92,891.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
278,675 ÷ 5 = 55,735 - No remainder! 5 is one of the factors!
55,735 ÷ 5 = 11,147 - No remainder! 5 is one of the factors!
11,147 ÷ 5 = 2,229.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
11,147 ÷ 7 = 1,592.4286 - This has a remainder. 7 is not a factor.
11,147 ÷ 11 = 1,013.3636 - This has a remainder. 11 is not a factor.
11,147 ÷ 13 = 857.4615 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,147 ÷ 71 = 157 - No remainder! 71 is one of the factors!
157 ÷ 71 = 2.2113 - There is a remainder. We can't divide by 71 evenly anymore. Let's try the next prime number
157 ÷ 73 = 2.1507 - This has a remainder. 73 is not a factor.
157 ÷ 79 = 1.9873 - This has a remainder. 79 is not a factor.
157 ÷ 83 = 1.8916 - This has a remainder. 83 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
157 ÷ 157 = 1 - No remainder! 157 is one of the factors!

The orange divisor(s) above are the prime factors of the number 214,022,400. 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 3 x 5 x 5 x 71 x 157 = 214,022,400. It can also be written in exponential form as 28 x 31 x 52 x 711 x 1571.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 214,022,400.

214,022,400
Factor Arrows
2107,011,200
Factor Arrows
253,505,600
Factor Arrows
226,752,800
Factor Arrows
213,376,400
Factor Arrows
26,688,200
Factor Arrows
23,344,100
Factor Arrows
21,672,050
Factor Arrows
2836,025
Factor Arrows
3278,675
Factor Arrows
555,735
Factor Arrows
511,147
Factor Arrows
71157

More Prime Factorization Examples

214,022,398214,022,399214,022,401214,022,402
21 x 107,011,1991214,022,3991171 x 12,589,553121 x 111 x 9,728,2911

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