Q: What is the prime factorization of the number 21,010,368?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 17 x 41 x 157
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 17, 41, 157 }
  • Written in exponential form: 26 x 31 x 171 x 411 x 1571

Why is the prime factorization of 21,010,368 written as 26 x 31 x 171 x 411 x 1571?

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 21,010,368

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 21,010,368 by 2

21,010,368 ÷ 2 = 10,505,184 - No remainder! 2 is one of the factors!
10,505,184 ÷ 2 = 5,252,592 - No remainder! 2 is one of the factors!
5,252,592 ÷ 2 = 2,626,296 - No remainder! 2 is one of the factors!
2,626,296 ÷ 2 = 1,313,148 - No remainder! 2 is one of the factors!
1,313,148 ÷ 2 = 656,574 - No remainder! 2 is one of the factors!
656,574 ÷ 2 = 328,287 - No remainder! 2 is one of the factors!
328,287 ÷ 2 = 164,143.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
328,287 ÷ 3 = 109,429 - No remainder! 3 is one of the factors!
109,429 ÷ 3 = 36,476.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
109,429 ÷ 5 = 21,885.8 - This has a remainder. 5 is not a factor.
109,429 ÷ 7 = 15,632.7143 - This has a remainder. 7 is not a factor.
109,429 ÷ 11 = 9,948.0909 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
109,429 ÷ 17 = 6,437 - No remainder! 17 is one of the factors!
6,437 ÷ 17 = 378.6471 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
6,437 ÷ 19 = 338.7895 - This has a remainder. 19 is not a factor.
6,437 ÷ 23 = 279.8696 - This has a remainder. 23 is not a factor.
6,437 ÷ 29 = 221.9655 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
6,437 ÷ 41 = 157 - No remainder! 41 is one of the factors!
157 ÷ 41 = 3.8293 - There is a remainder. We can't divide by 41 evenly anymore. Let's try the next prime number
157 ÷ 43 = 3.6512 - This has a remainder. 43 is not a factor.
157 ÷ 47 = 3.3404 - This has a remainder. 47 is not a factor.
157 ÷ 53 = 2.9623 - This has a remainder. 53 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
157 ÷ 157 = 1 - No remainder! 157 is one of the factors!

The orange divisor(s) above are the prime factors of the number 21,010,368. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 17 x 41 x 157 = 21,010,368. It can also be written in exponential form as 26 x 31 x 171 x 411 x 1571.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 21,010,368.

21,010,368
Factor Arrows
210,505,184
Factor Arrows
25,252,592
Factor Arrows
22,626,296
Factor Arrows
21,313,148
Factor Arrows
2656,574
Factor Arrows
2328,287
Factor Arrows
3109,429
Factor Arrows
176,437
Factor Arrows
41157

More Prime Factorization Examples

21,010,36621,010,36721,010,36921,010,370
21 x 131 x 531 x 791 x 193172 x 5211 x 82318871 x 23,687121 x 51 x 2811 x 7,4771

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