Q: What is the prime factorization of the number 23,403,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 5 x 13 x 13 x 577
    • or also written as { 2, 2, 2, 2, 3, 5, 13, 13, 577 }
  • Written in exponential form: 24 x 31 x 51 x 132 x 5771

Why is the prime factorization of 23,403,120 written as 24 x 31 x 51 x 132 x 5771?

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 23,403,120

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 23,403,120 by 2

23,403,120 ÷ 2 = 11,701,560 - No remainder! 2 is one of the factors!
11,701,560 ÷ 2 = 5,850,780 - No remainder! 2 is one of the factors!
5,850,780 ÷ 2 = 2,925,390 - No remainder! 2 is one of the factors!
2,925,390 ÷ 2 = 1,462,695 - No remainder! 2 is one of the factors!
1,462,695 ÷ 2 = 731,347.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,462,695 ÷ 3 = 487,565 - No remainder! 3 is one of the factors!
487,565 ÷ 3 = 162,521.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
487,565 ÷ 5 = 97,513 - No remainder! 5 is one of the factors!
97,513 ÷ 5 = 19,502.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
97,513 ÷ 7 = 13,930.4286 - This has a remainder. 7 is not a factor.
97,513 ÷ 11 = 8,864.8182 - This has a remainder. 11 is not a factor.
97,513 ÷ 13 = 7,501 - No remainder! 13 is one of the factors!
7,501 ÷ 13 = 577 - No remainder! 13 is one of the factors!
577 ÷ 13 = 44.3846 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
577 ÷ 17 = 33.9412 - This has a remainder. 17 is not a factor.
577 ÷ 19 = 30.3684 - This has a remainder. 19 is not a factor.
577 ÷ 23 = 25.087 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
577 ÷ 577 = 1 - No remainder! 577 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,403,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 5 x 13 x 13 x 577 = 23,403,120. It can also be written in exponential form as 24 x 31 x 51 x 132 x 5771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,403,120.

23,403,120
Factor Arrows
211,701,560
Factor Arrows
25,850,780
Factor Arrows
22,925,390
Factor Arrows
21,462,695
Factor Arrows
3487,565
Factor Arrows
597,513
Factor Arrows
137,501
Factor Arrows
13577

More Prime Factorization Examples

23,403,11823,403,11923,403,12123,403,122
21 x 171 x 791 x 8,71311811 x 2391 x 541171 x 231 x 145,361121 x 3,0371 x 3,8531

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