Q: What is the prime factorization of the number 12,950,400?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 19 x 71
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 19, 71 }
  • Written in exponential form: 27 x 31 x 52 x 191 x 711

Why is the prime factorization of 12,950,400 written as 27 x 31 x 52 x 191 x 711?

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 12,950,400

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 12,950,400 by 2

12,950,400 ÷ 2 = 6,475,200 - No remainder! 2 is one of the factors!
6,475,200 ÷ 2 = 3,237,600 - No remainder! 2 is one of the factors!
3,237,600 ÷ 2 = 1,618,800 - No remainder! 2 is one of the factors!
1,618,800 ÷ 2 = 809,400 - No remainder! 2 is one of the factors!
809,400 ÷ 2 = 404,700 - No remainder! 2 is one of the factors!
404,700 ÷ 2 = 202,350 - No remainder! 2 is one of the factors!
202,350 ÷ 2 = 101,175 - No remainder! 2 is one of the factors!
101,175 ÷ 2 = 50,587.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
101,175 ÷ 3 = 33,725 - No remainder! 3 is one of the factors!
33,725 ÷ 3 = 11,241.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
33,725 ÷ 5 = 6,745 - No remainder! 5 is one of the factors!
6,745 ÷ 5 = 1,349 - No remainder! 5 is one of the factors!
1,349 ÷ 5 = 269.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,349 ÷ 7 = 192.7143 - This has a remainder. 7 is not a factor.
1,349 ÷ 11 = 122.6364 - This has a remainder. 11 is not a factor.
1,349 ÷ 13 = 103.7692 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,349 ÷ 19 = 71 - No remainder! 19 is one of the factors!
71 ÷ 19 = 3.7368 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
71 ÷ 23 = 3.087 - This has a remainder. 23 is not a factor.
71 ÷ 29 = 2.4483 - This has a remainder. 29 is not a factor.
71 ÷ 31 = 2.2903 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
71 ÷ 71 = 1 - No remainder! 71 is one of the factors!

The orange divisor(s) above are the prime factors of the number 12,950,400. 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 3 x 5 x 5 x 19 x 71 = 12,950,400. It can also be written in exponential form as 27 x 31 x 52 x 191 x 711.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 12,950,400.

12,950,400
Factor Arrows
26,475,200
Factor Arrows
23,237,600
Factor Arrows
21,618,800
Factor Arrows
2809,400
Factor Arrows
2404,700
Factor Arrows
2202,350
Factor Arrows
2101,175
Factor Arrows
333,725
Factor Arrows
56,745
Factor Arrows
51,349
Factor Arrows
1971

More Prime Factorization Examples

12,950,39812,950,39912,950,40112,950,402
21 x 1311 x 49,429171 x 111 x 1091 x 1,54313171 x 40,853121 x 2,3711 x 2,7311

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