Q: What is the prime factorization of the number 21,294?

 A:
  • The prime factors are: 2 x 3 x 3 x 7 x 13 x 13
    • or also written as { 2, 3, 3, 7, 13, 13 }
  • Written in exponential form: 21 x 32 x 71 x 132

Why is the prime factorization of 21,294 written as 21 x 32 x 71 x 132?

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 21,294

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 21,294 by 2

21,294 ÷ 2 = 10,647 - No remainder! 2 is one of the factors!
10,647 ÷ 2 = 5,323.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,647 ÷ 3 = 3,549 - No remainder! 3 is one of the factors!
3,549 ÷ 3 = 1,183 - No remainder! 3 is one of the factors!
1,183 ÷ 3 = 394.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,183 ÷ 5 = 236.6 - This has a remainder. 5 is not a factor.
1,183 ÷ 7 = 169 - No remainder! 7 is one of the factors!
169 ÷ 7 = 24.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
169 ÷ 11 = 15.3636 - This has a remainder. 11 is not a factor.
169 ÷ 13 = 13 - No remainder! 13 is one of the factors!
13 ÷ 13 = 1 - No remainder! 13 is one of the factors!

The orange divisor(s) above are the prime factors of the number 21,294. If we put all of it together we have the factors 2 x 3 x 3 x 7 x 13 x 13 = 21,294. It can also be written in exponential form as 21 x 32 x 71 x 132.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 21,294.

21,294
Factor Arrows
210,647
Factor Arrows
33,549
Factor Arrows
31,183
Factor Arrows
7169
Factor Arrows
1313

More Prime Factorization Examples

21,29221,29321,29521,296
22 x 5,32311071 x 199151 x 4,259124 x 113

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