Q: What is the prime factorization of the number 23,040?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5 }
  • Written in exponential form: 29 x 32 x 51

Why is the prime factorization of 23,040 written as 29 x 32 x 51?

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 23,040

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 23,040 by 2

23,040 ÷ 2 = 11,520 - No remainder! 2 is one of the factors!
11,520 ÷ 2 = 5,760 - No remainder! 2 is one of the factors!
5,760 ÷ 2 = 2,880 - No remainder! 2 is one of the factors!
2,880 ÷ 2 = 1,440 - No remainder! 2 is one of the factors!
1,440 ÷ 2 = 720 - No remainder! 2 is one of the factors!
720 ÷ 2 = 360 - No remainder! 2 is one of the factors!
360 ÷ 2 = 180 - No remainder! 2 is one of the factors!
180 ÷ 2 = 90 - No remainder! 2 is one of the factors!
90 ÷ 2 = 45 - No remainder! 2 is one of the factors!
45 ÷ 2 = 22.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
45 ÷ 3 = 15 - No remainder! 3 is one of the factors!
15 ÷ 3 = 5 - No remainder! 3 is one of the factors!
5 ÷ 3 = 1.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
5 ÷ 5 = 1 - No remainder! 5 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,040. 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 3 x 3 x 5 = 23,040. It can also be written in exponential form as 29 x 32 x 51.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,040.

23,040
Factor Arrows
211,520
Factor Arrows
25,760
Factor Arrows
22,880
Factor Arrows
21,440
Factor Arrows
2720
Factor Arrows
2360
Factor Arrows
2180
Factor Arrows
290
Factor Arrows
245
Factor Arrows
315
Factor Arrows
35

More Prime Factorization Examples

23,03823,03923,04123,042
21 x 11,519123,039123,041121 x 411 x 2811

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