Q: What is the prime factorization of the number 13,230,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 7 x 7
    • or also written as { 2, 2, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7 }
  • Written in exponential form: 24 x 33 x 54 x 72

Why is the prime factorization of 13,230,000 written as 24 x 33 x 54 x 72?

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 13,230,000

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 13,230,000 by 2

13,230,000 ÷ 2 = 6,615,000 - No remainder! 2 is one of the factors!
6,615,000 ÷ 2 = 3,307,500 - No remainder! 2 is one of the factors!
3,307,500 ÷ 2 = 1,653,750 - No remainder! 2 is one of the factors!
1,653,750 ÷ 2 = 826,875 - No remainder! 2 is one of the factors!
826,875 ÷ 2 = 413,437.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
826,875 ÷ 3 = 275,625 - No remainder! 3 is one of the factors!
275,625 ÷ 3 = 91,875 - No remainder! 3 is one of the factors!
91,875 ÷ 3 = 30,625 - No remainder! 3 is one of the factors!
30,625 ÷ 3 = 10,208.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
30,625 ÷ 5 = 6,125 - No remainder! 5 is one of the factors!
6,125 ÷ 5 = 1,225 - No remainder! 5 is one of the factors!
1,225 ÷ 5 = 245 - No remainder! 5 is one of the factors!
245 ÷ 5 = 49 - No remainder! 5 is one of the factors!
49 ÷ 5 = 9.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
49 ÷ 7 = 7 - No remainder! 7 is one of the factors!
7 ÷ 7 = 1 - No remainder! 7 is one of the factors!

The orange divisor(s) above are the prime factors of the number 13,230,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 7 x 7 = 13,230,000. It can also be written in exponential form as 24 x 33 x 54 x 72.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 13,230,000.

13,230,000
Factor Arrows
26,615,000
Factor Arrows
23,307,500
Factor Arrows
21,653,750
Factor Arrows
2826,875
Factor Arrows
3275,625
Factor Arrows
391,875
Factor Arrows
330,625
Factor Arrows
56,125
Factor Arrows
51,225
Factor Arrows
5245
Factor Arrows
549
Factor Arrows
77

More Prime Factorization Examples

13,229,99813,229,99913,230,00113,230,002
21 x 711 x 93,169113,229,999113,230,001121 x 1731 x 38,2371

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