Q: What is the prime factorization of the number 13,634,352?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 3 x 37 x 853
    • or also written as { 2, 2, 2, 2, 3, 3, 3, 37, 853 }
  • Written in exponential form: 24 x 33 x 371 x 8531

Why is the prime factorization of 13,634,352 written as 24 x 33 x 371 x 8531?

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 13,634,352

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 13,634,352 by 2

13,634,352 ÷ 2 = 6,817,176 - No remainder! 2 is one of the factors!
6,817,176 ÷ 2 = 3,408,588 - No remainder! 2 is one of the factors!
3,408,588 ÷ 2 = 1,704,294 - No remainder! 2 is one of the factors!
1,704,294 ÷ 2 = 852,147 - No remainder! 2 is one of the factors!
852,147 ÷ 2 = 426,073.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
852,147 ÷ 3 = 284,049 - No remainder! 3 is one of the factors!
284,049 ÷ 3 = 94,683 - No remainder! 3 is one of the factors!
94,683 ÷ 3 = 31,561 - No remainder! 3 is one of the factors!
31,561 ÷ 3 = 10,520.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
31,561 ÷ 5 = 6,312.2 - This has a remainder. 5 is not a factor.
31,561 ÷ 7 = 4,508.7143 - This has a remainder. 7 is not a factor.
31,561 ÷ 11 = 2,869.1818 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
31,561 ÷ 37 = 853 - No remainder! 37 is one of the factors!
853 ÷ 37 = 23.0541 - There is a remainder. We can't divide by 37 evenly anymore. Let's try the next prime number
853 ÷ 41 = 20.8049 - This has a remainder. 41 is not a factor.
853 ÷ 43 = 19.8372 - This has a remainder. 43 is not a factor.
853 ÷ 47 = 18.1489 - This has a remainder. 47 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
853 ÷ 853 = 1 - No remainder! 853 is one of the factors!

The orange divisor(s) above are the prime factors of the number 13,634,352. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 3 x 37 x 853 = 13,634,352. It can also be written in exponential form as 24 x 33 x 371 x 8531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 13,634,352.

13,634,352
Factor Arrows
26,817,176
Factor Arrows
23,408,588
Factor Arrows
21,704,294
Factor Arrows
2852,147
Factor Arrows
3284,049
Factor Arrows
394,683
Factor Arrows
331,561
Factor Arrows
37853

More Prime Factorization Examples

13,634,35013,634,35113,634,35313,634,354
21 x 52 x 291 x 9,40311571 x 86,843113,634,353121 x 231 x 431 x 611 x 1131

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