Q: What is the prime factorization of the number 13,331,402?

 A:
  • The prime factors are: 2 x 7 x 211 x 4,513
    • or also written as { 2, 7, 211, 4,513 }
  • Written in exponential form: 21 x 71 x 2111 x 4,5131

Why is the prime factorization of 13,331,402 written as 21 x 71 x 2111 x 4,5131?

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 13,331,402

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 13,331,402 by 2

13,331,402 ÷ 2 = 6,665,701 - No remainder! 2 is one of the factors!
6,665,701 ÷ 2 = 3,332,850.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
6,665,701 ÷ 3 = 2,221,900.3333 - This has a remainder. 3 is not a factor.
6,665,701 ÷ 5 = 1,333,140.2 - This has a remainder. 5 is not a factor.
6,665,701 ÷ 7 = 952,243 - No remainder! 7 is one of the factors!
952,243 ÷ 7 = 136,034.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
952,243 ÷ 11 = 86,567.5455 - This has a remainder. 11 is not a factor.
952,243 ÷ 13 = 73,249.4615 - This has a remainder. 13 is not a factor.
952,243 ÷ 17 = 56,014.2941 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
952,243 ÷ 211 = 4,513 - No remainder! 211 is one of the factors!
4,513 ÷ 211 = 21.3886 - There is a remainder. We can't divide by 211 evenly anymore. Let's try the next prime number
4,513 ÷ 223 = 20.2377 - This has a remainder. 223 is not a factor.
4,513 ÷ 227 = 19.8811 - This has a remainder. 227 is not a factor.
4,513 ÷ 229 = 19.7074 - This has a remainder. 229 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,513 ÷ 4,513 = 1 - No remainder! 4,513 is one of the factors!

The orange divisor(s) above are the prime factors of the number 13,331,402. If we put all of it together we have the factors 2 x 7 x 211 x 4,513 = 13,331,402. It can also be written in exponential form as 21 x 71 x 2111 x 4,5131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 13,331,402.

13,331,402
Factor Arrows
26,665,701
Factor Arrows
7952,243
Factor Arrows
2114,513

More Prime Factorization Examples

13,331,40013,331,40113,331,40313,331,404
23 x 31 x 52 x 171 x 1,30711131 x 117,977132 x 6191 x 2,393122 x 591 x 56,4891

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