Q: What is the prime factorization of the number 1,215,168?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 6,329
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 6,329 }
  • Written in exponential form: 26 x 31 x 6,3291

Why is the prime factorization of 1,215,168 written as 26 x 31 x 6,3291?

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 1,215,168

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 1,215,168 by 2

1,215,168 ÷ 2 = 607,584 - No remainder! 2 is one of the factors!
607,584 ÷ 2 = 303,792 - No remainder! 2 is one of the factors!
303,792 ÷ 2 = 151,896 - No remainder! 2 is one of the factors!
151,896 ÷ 2 = 75,948 - No remainder! 2 is one of the factors!
75,948 ÷ 2 = 37,974 - No remainder! 2 is one of the factors!
37,974 ÷ 2 = 18,987 - No remainder! 2 is one of the factors!
18,987 ÷ 2 = 9,493.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
18,987 ÷ 3 = 6,329 - No remainder! 3 is one of the factors!
6,329 ÷ 3 = 2,109.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,329 ÷ 5 = 1,265.8 - This has a remainder. 5 is not a factor.
6,329 ÷ 7 = 904.1429 - This has a remainder. 7 is not a factor.
6,329 ÷ 11 = 575.3636 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
6,329 ÷ 6,329 = 1 - No remainder! 6,329 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,215,168. 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 6,329 = 1,215,168. It can also be written in exponential form as 26 x 31 x 6,3291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,215,168.

1,215,168
Factor Arrows
2607,584
Factor Arrows
2303,792
Factor Arrows
2151,896
Factor Arrows
275,948
Factor Arrows
237,974
Factor Arrows
218,987
Factor Arrows
36,329

More Prime Factorization Examples

1,215,1661,215,1671,215,1691,215,170
21 x 607,58311,215,1671311 x 39,199121 x 51 x 111 x 11,0471

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