Q: What is the prime factorization of the number 25,536,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 7 x 19
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 5, 7, 19 }
  • Written in exponential form: 29 x 31 x 53 x 71 x 191

Why is the prime factorization of 25,536,000 written as 29 x 31 x 53 x 71 x 191?

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 25,536,000

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 25,536,000 by 2

25,536,000 ÷ 2 = 12,768,000 - No remainder! 2 is one of the factors!
12,768,000 ÷ 2 = 6,384,000 - No remainder! 2 is one of the factors!
6,384,000 ÷ 2 = 3,192,000 - No remainder! 2 is one of the factors!
3,192,000 ÷ 2 = 1,596,000 - No remainder! 2 is one of the factors!
1,596,000 ÷ 2 = 798,000 - No remainder! 2 is one of the factors!
798,000 ÷ 2 = 399,000 - No remainder! 2 is one of the factors!
399,000 ÷ 2 = 199,500 - No remainder! 2 is one of the factors!
199,500 ÷ 2 = 99,750 - No remainder! 2 is one of the factors!
99,750 ÷ 2 = 49,875 - No remainder! 2 is one of the factors!
49,875 ÷ 2 = 24,937.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
49,875 ÷ 3 = 16,625 - No remainder! 3 is one of the factors!
16,625 ÷ 3 = 5,541.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
16,625 ÷ 5 = 3,325 - No remainder! 5 is one of the factors!
3,325 ÷ 5 = 665 - No remainder! 5 is one of the factors!
665 ÷ 5 = 133 - No remainder! 5 is one of the factors!
133 ÷ 5 = 26.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
133 ÷ 7 = 19 - No remainder! 7 is one of the factors!
19 ÷ 7 = 2.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
19 ÷ 11 = 1.7273 - This has a remainder. 11 is not a factor.
19 ÷ 13 = 1.4615 - This has a remainder. 13 is not a factor.
19 ÷ 17 = 1.1176 - This has a remainder. 17 is not a factor.
19 ÷ 19 = 1 - No remainder! 19 is one of the factors!

The orange divisor(s) above are the prime factors of the number 25,536,000. 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 3 x 5 x 5 x 5 x 7 x 19 = 25,536,000. It can also be written in exponential form as 29 x 31 x 53 x 71 x 191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 25,536,000.

25,536,000
Factor Arrows
212,768,000
Factor Arrows
26,384,000
Factor Arrows
23,192,000
Factor Arrows
21,596,000
Factor Arrows
2798,000
Factor Arrows
2399,000
Factor Arrows
2199,500
Factor Arrows
299,750
Factor Arrows
249,875
Factor Arrows
316,625
Factor Arrows
53,325
Factor Arrows
5665
Factor Arrows
5133
Factor Arrows
719

More Prime Factorization Examples

25,535,99825,535,99925,536,00125,536,002
21 x 5231 x 24,413125,535,999125,536,001121 x 311 x 711 x 5,8011

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