Q: What is the prime factorization of the number 14,516,334?

 A:
  • The prime factors are: 2 x 3 x 3 x 3 x 3 x 3 x 7 x 17 x 251
    • or also written as { 2, 3, 3, 3, 3, 3, 7, 17, 251 }
  • Written in exponential form: 21 x 35 x 71 x 171 x 2511

Why is the prime factorization of 14,516,334 written as 21 x 35 x 71 x 171 x 2511?

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 14,516,334

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 14,516,334 by 2

14,516,334 ÷ 2 = 7,258,167 - No remainder! 2 is one of the factors!
7,258,167 ÷ 2 = 3,629,083.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
7,258,167 ÷ 3 = 2,419,389 - No remainder! 3 is one of the factors!
2,419,389 ÷ 3 = 806,463 - No remainder! 3 is one of the factors!
806,463 ÷ 3 = 268,821 - No remainder! 3 is one of the factors!
268,821 ÷ 3 = 89,607 - No remainder! 3 is one of the factors!
89,607 ÷ 3 = 29,869 - No remainder! 3 is one of the factors!
29,869 ÷ 3 = 9,956.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
29,869 ÷ 5 = 5,973.8 - This has a remainder. 5 is not a factor.
29,869 ÷ 7 = 4,267 - No remainder! 7 is one of the factors!
4,267 ÷ 7 = 609.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
4,267 ÷ 11 = 387.9091 - This has a remainder. 11 is not a factor.
4,267 ÷ 13 = 328.2308 - This has a remainder. 13 is not a factor.
4,267 ÷ 17 = 251 - No remainder! 17 is one of the factors!
251 ÷ 17 = 14.7647 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
251 ÷ 19 = 13.2105 - This has a remainder. 19 is not a factor.
251 ÷ 23 = 10.913 - This has a remainder. 23 is not a factor.
251 ÷ 29 = 8.6552 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
251 ÷ 251 = 1 - No remainder! 251 is one of the factors!

The orange divisor(s) above are the prime factors of the number 14,516,334. If we put all of it together we have the factors 2 x 3 x 3 x 3 x 3 x 3 x 7 x 17 x 251 = 14,516,334. It can also be written in exponential form as 21 x 35 x 71 x 171 x 2511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 14,516,334.

14,516,334
Factor Arrows
27,258,167
Factor Arrows
32,419,389
Factor Arrows
3806,463
Factor Arrows
3268,821
Factor Arrows
389,607
Factor Arrows
329,869
Factor Arrows
74,267
Factor Arrows
17251

More Prime Factorization Examples

14,516,33214,516,33314,516,33514,516,336
22 x 7331 x 4,9511131 x 1,116,641151 x 231 x 126,229124 x 3311 x 2,7411

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