Q: What is the prime factorization of the number 23,737,308?

 A:
  • The prime factors are: 2 x 2 x 3 x 7 x 19 x 107 x 139
    • or also written as { 2, 2, 3, 7, 19, 107, 139 }
  • Written in exponential form: 22 x 31 x 71 x 191 x 1071 x 1391

Why is the prime factorization of 23,737,308 written as 22 x 31 x 71 x 191 x 1071 x 1391?

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 23,737,308

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 23,737,308 by 2

23,737,308 ÷ 2 = 11,868,654 - No remainder! 2 is one of the factors!
11,868,654 ÷ 2 = 5,934,327 - No remainder! 2 is one of the factors!
5,934,327 ÷ 2 = 2,967,163.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
5,934,327 ÷ 3 = 1,978,109 - No remainder! 3 is one of the factors!
1,978,109 ÷ 3 = 659,369.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,978,109 ÷ 5 = 395,621.8 - This has a remainder. 5 is not a factor.
1,978,109 ÷ 7 = 282,587 - No remainder! 7 is one of the factors!
282,587 ÷ 7 = 40,369.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
282,587 ÷ 11 = 25,689.7273 - This has a remainder. 11 is not a factor.
282,587 ÷ 13 = 21,737.4615 - This has a remainder. 13 is not a factor.
282,587 ÷ 17 = 16,622.7647 - This has a remainder. 17 is not a factor.
282,587 ÷ 19 = 14,873 - No remainder! 19 is one of the factors!
14,873 ÷ 19 = 782.7895 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
14,873 ÷ 23 = 646.6522 - This has a remainder. 23 is not a factor.
14,873 ÷ 29 = 512.8621 - This has a remainder. 29 is not a factor.
14,873 ÷ 31 = 479.7742 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
14,873 ÷ 107 = 139 - No remainder! 107 is one of the factors!
139 ÷ 107 = 1.2991 - There is a remainder. We can't divide by 107 evenly anymore. Let's try the next prime number
139 ÷ 109 = 1.2752 - This has a remainder. 109 is not a factor.
139 ÷ 113 = 1.2301 - This has a remainder. 113 is not a factor.
139 ÷ 127 = 1.0945 - This has a remainder. 127 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
139 ÷ 139 = 1 - No remainder! 139 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,737,308. If we put all of it together we have the factors 2 x 2 x 3 x 7 x 19 x 107 x 139 = 23,737,308. It can also be written in exponential form as 22 x 31 x 71 x 191 x 1071 x 1391.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,737,308.

23,737,308
Factor Arrows
211,868,654
Factor Arrows
25,934,327
Factor Arrows
31,978,109
Factor Arrows
7282,587
Factor Arrows
1914,873
Factor Arrows
107139

More Prime Factorization Examples

23,737,30623,737,30723,737,30923,737,310
21 x 11,868,6531111 x 1,0331 x 2,08914,0991 x 5,791121 x 51 x 2,373,7311

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