Q: What is the prime factorization of the number 24,201,500?

 A:
  • The prime factors are: 2 x 2 x 5 x 5 x 5 x 97 x 499
    • or also written as { 2, 2, 5, 5, 5, 97, 499 }
  • Written in exponential form: 22 x 53 x 971 x 4991

Why is the prime factorization of 24,201,500 written as 22 x 53 x 971 x 4991?

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 24,201,500

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 24,201,500 by 2

24,201,500 ÷ 2 = 12,100,750 - No remainder! 2 is one of the factors!
12,100,750 ÷ 2 = 6,050,375 - No remainder! 2 is one of the factors!
6,050,375 ÷ 2 = 3,025,187.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,050,375 ÷ 3 = 2,016,791.6667 - This has a remainder. 3 is not a factor.
6,050,375 ÷ 5 = 1,210,075 - No remainder! 5 is one of the factors!
1,210,075 ÷ 5 = 242,015 - No remainder! 5 is one of the factors!
242,015 ÷ 5 = 48,403 - No remainder! 5 is one of the factors!
48,403 ÷ 5 = 9,680.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
48,403 ÷ 7 = 6,914.7143 - This has a remainder. 7 is not a factor.
48,403 ÷ 11 = 4,400.2727 - This has a remainder. 11 is not a factor.
48,403 ÷ 13 = 3,723.3077 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
48,403 ÷ 97 = 499 - No remainder! 97 is one of the factors!
499 ÷ 97 = 5.1443 - There is a remainder. We can't divide by 97 evenly anymore. Let's try the next prime number
499 ÷ 101 = 4.9406 - This has a remainder. 101 is not a factor.
499 ÷ 103 = 4.8447 - This has a remainder. 103 is not a factor.
499 ÷ 107 = 4.6636 - This has a remainder. 107 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
499 ÷ 499 = 1 - No remainder! 499 is one of the factors!

The orange divisor(s) above are the prime factors of the number 24,201,500. If we put all of it together we have the factors 2 x 2 x 5 x 5 x 5 x 97 x 499 = 24,201,500. It can also be written in exponential form as 22 x 53 x 971 x 4991.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 24,201,500.

24,201,500
Factor Arrows
212,100,750
Factor Arrows
26,050,375
Factor Arrows
51,210,075
Factor Arrows
5242,015
Factor Arrows
548,403
Factor Arrows
97499

More Prime Factorization Examples

24,201,49824,201,49924,201,50124,201,502
21 x 31 x 1031 x 39,161171 x 3,457,357131 x 1271 x 63,521121 x 131 x 930,8271

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