Q: What is the prime factorization of the number 23,200,320?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 11 x 13 x 13 x 13
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 5, 11, 13, 13, 13 }
  • Written in exponential form: 26 x 31 x 51 x 111 x 133

Why is the prime factorization of 23,200,320 written as 26 x 31 x 51 x 111 x 133?

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 23,200,320

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 23,200,320 by 2

23,200,320 ÷ 2 = 11,600,160 - No remainder! 2 is one of the factors!
11,600,160 ÷ 2 = 5,800,080 - No remainder! 2 is one of the factors!
5,800,080 ÷ 2 = 2,900,040 - No remainder! 2 is one of the factors!
2,900,040 ÷ 2 = 1,450,020 - No remainder! 2 is one of the factors!
1,450,020 ÷ 2 = 725,010 - No remainder! 2 is one of the factors!
725,010 ÷ 2 = 362,505 - No remainder! 2 is one of the factors!
362,505 ÷ 2 = 181,252.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
362,505 ÷ 3 = 120,835 - No remainder! 3 is one of the factors!
120,835 ÷ 3 = 40,278.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
120,835 ÷ 5 = 24,167 - No remainder! 5 is one of the factors!
24,167 ÷ 5 = 4,833.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
24,167 ÷ 7 = 3,452.4286 - This has a remainder. 7 is not a factor.
24,167 ÷ 11 = 2,197 - No remainder! 11 is one of the factors!
2,197 ÷ 11 = 199.7273 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
2,197 ÷ 13 = 169 - No remainder! 13 is one of the factors!
169 ÷ 13 = 13 - No remainder! 13 is one of the factors!
13 ÷ 13 = 1 - No remainder! 13 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,200,320. 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 11 x 13 x 13 x 13 = 23,200,320. It can also be written in exponential form as 26 x 31 x 51 x 111 x 133.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,200,320.

23,200,320
Factor Arrows
211,600,160
Factor Arrows
25,800,080
Factor Arrows
22,900,040
Factor Arrows
21,450,020
Factor Arrows
2725,010
Factor Arrows
2362,505
Factor Arrows
3120,835
Factor Arrows
524,167
Factor Arrows
112,197
Factor Arrows
13169
Factor Arrows
1313

More Prime Factorization Examples

23,200,31823,200,31923,200,32123,200,322
21 x 11,600,1591291 x 800,011123,200,321121 x 11,600,1611

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