Q: What is the prime factorization of the number 12,423,216?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 13 x 43 x 463
    • or also written as { 2, 2, 2, 2, 3, 13, 43, 463 }
  • Written in exponential form: 24 x 31 x 131 x 431 x 4631

Why is the prime factorization of 12,423,216 written as 24 x 31 x 131 x 431 x 4631?

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 12,423,216

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 12,423,216 by 2

12,423,216 ÷ 2 = 6,211,608 - No remainder! 2 is one of the factors!
6,211,608 ÷ 2 = 3,105,804 - No remainder! 2 is one of the factors!
3,105,804 ÷ 2 = 1,552,902 - No remainder! 2 is one of the factors!
1,552,902 ÷ 2 = 776,451 - No remainder! 2 is one of the factors!
776,451 ÷ 2 = 388,225.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
776,451 ÷ 3 = 258,817 - No remainder! 3 is one of the factors!
258,817 ÷ 3 = 86,272.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
258,817 ÷ 5 = 51,763.4 - This has a remainder. 5 is not a factor.
258,817 ÷ 7 = 36,973.8571 - This has a remainder. 7 is not a factor.
258,817 ÷ 11 = 23,528.8182 - This has a remainder. 11 is not a factor.
258,817 ÷ 13 = 19,909 - No remainder! 13 is one of the factors!
19,909 ÷ 13 = 1,531.4615 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
19,909 ÷ 17 = 1,171.1176 - This has a remainder. 17 is not a factor.
19,909 ÷ 19 = 1,047.8421 - This has a remainder. 19 is not a factor.
19,909 ÷ 23 = 865.6087 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
19,909 ÷ 43 = 463 - No remainder! 43 is one of the factors!
463 ÷ 43 = 10.7674 - There is a remainder. We can't divide by 43 evenly anymore. Let's try the next prime number
463 ÷ 47 = 9.8511 - This has a remainder. 47 is not a factor.
463 ÷ 53 = 8.7358 - This has a remainder. 53 is not a factor.
463 ÷ 59 = 7.8475 - This has a remainder. 59 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
463 ÷ 463 = 1 - No remainder! 463 is one of the factors!

The orange divisor(s) above are the prime factors of the number 12,423,216. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 13 x 43 x 463 = 12,423,216. It can also be written in exponential form as 24 x 31 x 131 x 431 x 4631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 12,423,216.

12,423,216
Factor Arrows
26,211,608
Factor Arrows
23,105,804
Factor Arrows
21,552,902
Factor Arrows
2776,451
Factor Arrows
3258,817
Factor Arrows
1319,909
Factor Arrows
43463

More Prime Factorization Examples

12,423,21412,423,21512,423,21712,423,218
21 x 1971 x 31,531151 x 72 x 50,7071591 x 1931 x 1,091121 x 6,211,6091

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