Q: What is the prime factorization of the number 14,835,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 19 x 61
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 19, 61 }
  • Written in exponential form: 29 x 52 x 191 x 611

Why is the prime factorization of 14,835,200 written as 29 x 52 x 191 x 611?

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 14,835,200

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 14,835,200 by 2

14,835,200 ÷ 2 = 7,417,600 - No remainder! 2 is one of the factors!
7,417,600 ÷ 2 = 3,708,800 - No remainder! 2 is one of the factors!
3,708,800 ÷ 2 = 1,854,400 - No remainder! 2 is one of the factors!
1,854,400 ÷ 2 = 927,200 - No remainder! 2 is one of the factors!
927,200 ÷ 2 = 463,600 - No remainder! 2 is one of the factors!
463,600 ÷ 2 = 231,800 - No remainder! 2 is one of the factors!
231,800 ÷ 2 = 115,900 - No remainder! 2 is one of the factors!
115,900 ÷ 2 = 57,950 - No remainder! 2 is one of the factors!
57,950 ÷ 2 = 28,975 - No remainder! 2 is one of the factors!
28,975 ÷ 2 = 14,487.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
28,975 ÷ 3 = 9,658.3333 - This has a remainder. 3 is not a factor.
28,975 ÷ 5 = 5,795 - No remainder! 5 is one of the factors!
5,795 ÷ 5 = 1,159 - No remainder! 5 is one of the factors!
1,159 ÷ 5 = 231.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,159 ÷ 7 = 165.5714 - This has a remainder. 7 is not a factor.
1,159 ÷ 11 = 105.3636 - This has a remainder. 11 is not a factor.
1,159 ÷ 13 = 89.1538 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,159 ÷ 19 = 61 - No remainder! 19 is one of the factors!
61 ÷ 19 = 3.2105 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
61 ÷ 23 = 2.6522 - This has a remainder. 23 is not a factor.
61 ÷ 29 = 2.1034 - This has a remainder. 29 is not a factor.
61 ÷ 31 = 1.9677 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
61 ÷ 61 = 1 - No remainder! 61 is one of the factors!

The orange divisor(s) above are the prime factors of the number 14,835,200. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 19 x 61 = 14,835,200. It can also be written in exponential form as 29 x 52 x 191 x 611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 14,835,200.

14,835,200
Factor Arrows
27,417,600
Factor Arrows
23,708,800
Factor Arrows
21,854,400
Factor Arrows
2927,200
Factor Arrows
2463,600
Factor Arrows
2231,800
Factor Arrows
2115,900
Factor Arrows
257,950
Factor Arrows
228,975
Factor Arrows
55,795
Factor Arrows
51,159
Factor Arrows
1961

More Prime Factorization Examples

14,835,19814,835,19914,835,20114,835,202
21 x 31 x 71 x 4211 x 83912111 x 70,309131 x 9291 x 5,323121 x 2,2431 x 3,3071

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