Q: What is the prime factorization of the number 23,033,208?

 A:
  • The prime factors are: 2 x 2 x 2 x 3 x 11 x 43 x 2,029
    • or also written as { 2, 2, 2, 3, 11, 43, 2,029 }
  • Written in exponential form: 23 x 31 x 111 x 431 x 2,0291

Why is the prime factorization of 23,033,208 written as 23 x 31 x 111 x 431 x 2,0291?

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 23,033,208

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 23,033,208 by 2

23,033,208 ÷ 2 = 11,516,604 - No remainder! 2 is one of the factors!
11,516,604 ÷ 2 = 5,758,302 - No remainder! 2 is one of the factors!
5,758,302 ÷ 2 = 2,879,151 - No remainder! 2 is one of the factors!
2,879,151 ÷ 2 = 1,439,575.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,879,151 ÷ 3 = 959,717 - No remainder! 3 is one of the factors!
959,717 ÷ 3 = 319,905.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
959,717 ÷ 5 = 191,943.4 - This has a remainder. 5 is not a factor.
959,717 ÷ 7 = 137,102.4286 - This has a remainder. 7 is not a factor.
959,717 ÷ 11 = 87,247 - No remainder! 11 is one of the factors!
87,247 ÷ 11 = 7,931.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
87,247 ÷ 13 = 6,711.3077 - This has a remainder. 13 is not a factor.
87,247 ÷ 17 = 5,132.1765 - This has a remainder. 17 is not a factor.
87,247 ÷ 19 = 4,591.9474 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
87,247 ÷ 43 = 2,029 - No remainder! 43 is one of the factors!
2,029 ÷ 43 = 47.186 - There is a remainder. We can't divide by 43 evenly anymore. Let's try the next prime number
2,029 ÷ 47 = 43.1702 - This has a remainder. 47 is not a factor.
2,029 ÷ 53 = 38.283 - This has a remainder. 53 is not a factor.
2,029 ÷ 59 = 34.3898 - This has a remainder. 59 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,029 ÷ 2,029 = 1 - No remainder! 2,029 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,033,208. If we put all of it together we have the factors 2 x 2 x 2 x 3 x 11 x 43 x 2,029 = 23,033,208. It can also be written in exponential form as 23 x 31 x 111 x 431 x 2,0291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,033,208.

23,033,208
Factor Arrows
211,516,604
Factor Arrows
25,758,302
Factor Arrows
22,879,151
Factor Arrows
3959,717
Factor Arrows
1187,247
Factor Arrows
432,029

More Prime Factorization Examples

23,033,20623,033,20723,033,20923,033,210
21 x 71 x 191 x 1311 x 66111,4271 x 16,14114,1331 x 5,573121 x 51 x 2,303,3211

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