Q: What is the prime factorization of the number 23,514,435?

 A:
  • The prime factors are: 3 x 3 x 3 x 5 x 7 x 149 x 167
    • or also written as { 3, 3, 3, 5, 7, 149, 167 }
  • Written in exponential form: 33 x 51 x 71 x 1491 x 1671

Why is the prime factorization of 23,514,435 written as 33 x 51 x 71 x 1491 x 1671?

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 23,514,435

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 23,514,435 by 2

23,514,435 ÷ 2 = 11,757,217.5 - This has a remainder. Let's try another prime number.
23,514,435 ÷ 3 = 7,838,145 - No remainder! 3 is one of the factors!
7,838,145 ÷ 3 = 2,612,715 - No remainder! 3 is one of the factors!
2,612,715 ÷ 3 = 870,905 - No remainder! 3 is one of the factors!
870,905 ÷ 3 = 290,301.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
870,905 ÷ 5 = 174,181 - No remainder! 5 is one of the factors!
174,181 ÷ 5 = 34,836.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
174,181 ÷ 7 = 24,883 - No remainder! 7 is one of the factors!
24,883 ÷ 7 = 3,554.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
24,883 ÷ 11 = 2,262.0909 - This has a remainder. 11 is not a factor.
24,883 ÷ 13 = 1,914.0769 - This has a remainder. 13 is not a factor.
24,883 ÷ 17 = 1,463.7059 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,883 ÷ 149 = 167 - No remainder! 149 is one of the factors!
167 ÷ 149 = 1.1208 - There is a remainder. We can't divide by 149 evenly anymore. Let's try the next prime number
167 ÷ 151 = 1.106 - This has a remainder. 151 is not a factor.
167 ÷ 157 = 1.0637 - This has a remainder. 157 is not a factor.
167 ÷ 163 = 1.0245 - This has a remainder. 163 is not a factor.
167 ÷ 167 = 1 - No remainder! 167 is one of the factors!

The orange divisor(s) above are the prime factors of the number 23,514,435. If we put all of it together we have the factors 3 x 3 x 3 x 5 x 7 x 149 x 167 = 23,514,435. It can also be written in exponential form as 33 x 51 x 71 x 1491 x 1671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 23,514,435.

23,514,435
Factor Arrows
37,838,145
Factor Arrows
32,612,715
Factor Arrows
3870,905
Factor Arrows
5174,181
Factor Arrows
724,883
Factor Arrows
149167

More Prime Factorization Examples

23,514,43323,514,43423,514,43623,514,437
2511 x 93,683121 x 171 x 1811 x 3,821122 x 111 x 5471 x 97711991 x 118,1631

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