Q: What is the prime factorization of the number 6,213,120?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 809
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 809 }
  • Written in exponential form: 29 x 31 x 51 x 8091

Why is the prime factorization of 6,213,120 written as 29 x 31 x 51 x 8091?

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 6,213,120

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 6,213,120 by 2

6,213,120 ÷ 2 = 3,106,560 - No remainder! 2 is one of the factors!
3,106,560 ÷ 2 = 1,553,280 - No remainder! 2 is one of the factors!
1,553,280 ÷ 2 = 776,640 - No remainder! 2 is one of the factors!
776,640 ÷ 2 = 388,320 - No remainder! 2 is one of the factors!
388,320 ÷ 2 = 194,160 - No remainder! 2 is one of the factors!
194,160 ÷ 2 = 97,080 - No remainder! 2 is one of the factors!
97,080 ÷ 2 = 48,540 - No remainder! 2 is one of the factors!
48,540 ÷ 2 = 24,270 - No remainder! 2 is one of the factors!
24,270 ÷ 2 = 12,135 - No remainder! 2 is one of the factors!
12,135 ÷ 2 = 6,067.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
12,135 ÷ 3 = 4,045 - No remainder! 3 is one of the factors!
4,045 ÷ 3 = 1,348.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
4,045 ÷ 5 = 809 - No remainder! 5 is one of the factors!
809 ÷ 5 = 161.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
809 ÷ 7 = 115.5714 - This has a remainder. 7 is not a factor.
809 ÷ 11 = 73.5455 - This has a remainder. 11 is not a factor.
809 ÷ 13 = 62.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
809 ÷ 809 = 1 - No remainder! 809 is one of the factors!

The orange divisor(s) above are the prime factors of the number 6,213,120. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 809 = 6,213,120. It can also be written in exponential form as 29 x 31 x 51 x 8091.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 6,213,120.

6,213,120
Factor Arrows
23,106,560
Factor Arrows
21,553,280
Factor Arrows
2776,640
Factor Arrows
2388,320
Factor Arrows
2194,160
Factor Arrows
297,080
Factor Arrows
248,540
Factor Arrows
224,270
Factor Arrows
212,135
Factor Arrows
34,045
Factor Arrows
5809

More Prime Factorization Examples

6,213,1186,213,1196,213,1216,213,122
21 x 471 x 1571 x 4211111 x 4091 x 1,38116,213,121121 x 1731 x 17,9571

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