Q: What is the prime factorization of the number 25,324,515?

 A:
  • The prime factors are: 3 x 3 x 3 x 5 x 109 x 1,721
    • or also written as { 3, 3, 3, 5, 109, 1,721 }
  • Written in exponential form: 33 x 51 x 1091 x 1,7211

Why is the prime factorization of 25,324,515 written as 33 x 51 x 1091 x 1,7211?

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 25,324,515

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 25,324,515 by 2

25,324,515 ÷ 2 = 12,662,257.5 - This has a remainder. Let's try another prime number.
25,324,515 ÷ 3 = 8,441,505 - No remainder! 3 is one of the factors!
8,441,505 ÷ 3 = 2,813,835 - No remainder! 3 is one of the factors!
2,813,835 ÷ 3 = 937,945 - No remainder! 3 is one of the factors!
937,945 ÷ 3 = 312,648.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
937,945 ÷ 5 = 187,589 - No remainder! 5 is one of the factors!
187,589 ÷ 5 = 37,517.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
187,589 ÷ 7 = 26,798.4286 - This has a remainder. 7 is not a factor.
187,589 ÷ 11 = 17,053.5455 - This has a remainder. 11 is not a factor.
187,589 ÷ 13 = 14,429.9231 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
187,589 ÷ 109 = 1,721 - No remainder! 109 is one of the factors!
1,721 ÷ 109 = 15.789 - There is a remainder. We can't divide by 109 evenly anymore. Let's try the next prime number
1,721 ÷ 113 = 15.2301 - This has a remainder. 113 is not a factor.
1,721 ÷ 127 = 13.5512 - This has a remainder. 127 is not a factor.
1,721 ÷ 131 = 13.1374 - This has a remainder. 131 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,721 ÷ 1,721 = 1 - No remainder! 1,721 is one of the factors!

The orange divisor(s) above are the prime factors of the number 25,324,515. If we put all of it together we have the factors 3 x 3 x 3 x 5 x 109 x 1,721 = 25,324,515. It can also be written in exponential form as 33 x 51 x 1091 x 1,7211.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 25,324,515.

25,324,515
Factor Arrows
38,441,505
Factor Arrows
32,813,835
Factor Arrows
3937,945
Factor Arrows
5187,589
Factor Arrows
1091,721

More Prime Factorization Examples

25,324,51325,324,51425,324,51625,324,517
531 x 477,821121 x 12,662,257122 x 71 x 611 x 14,82713131 x 80,9091

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