Q: What is the prime factorization of the number 3,827,712?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 89
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 89 }
  • Written in exponential form: 211 x 31 x 71 x 891

Why is the prime factorization of 3,827,712 written as 211 x 31 x 71 x 891?

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 3,827,712

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 3,827,712 by 2

3,827,712 ÷ 2 = 1,913,856 - No remainder! 2 is one of the factors!
1,913,856 ÷ 2 = 956,928 - No remainder! 2 is one of the factors!
956,928 ÷ 2 = 478,464 - No remainder! 2 is one of the factors!
478,464 ÷ 2 = 239,232 - No remainder! 2 is one of the factors!
239,232 ÷ 2 = 119,616 - No remainder! 2 is one of the factors!
119,616 ÷ 2 = 59,808 - No remainder! 2 is one of the factors!
59,808 ÷ 2 = 29,904 - No remainder! 2 is one of the factors!
29,904 ÷ 2 = 14,952 - No remainder! 2 is one of the factors!
14,952 ÷ 2 = 7,476 - No remainder! 2 is one of the factors!
7,476 ÷ 2 = 3,738 - No remainder! 2 is one of the factors!
3,738 ÷ 2 = 1,869 - No remainder! 2 is one of the factors!
1,869 ÷ 2 = 934.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,869 ÷ 3 = 623 - No remainder! 3 is one of the factors!
623 ÷ 3 = 207.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
623 ÷ 5 = 124.6 - This has a remainder. 5 is not a factor.
623 ÷ 7 = 89 - No remainder! 7 is one of the factors!
89 ÷ 7 = 12.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
89 ÷ 11 = 8.0909 - This has a remainder. 11 is not a factor.
89 ÷ 13 = 6.8462 - This has a remainder. 13 is not a factor.
89 ÷ 17 = 5.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
89 ÷ 89 = 1 - No remainder! 89 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,827,712. 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 2 x 2 x 3 x 7 x 89 = 3,827,712. It can also be written in exponential form as 211 x 31 x 71 x 891.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,827,712.

3,827,712
Factor Arrows
21,913,856
Factor Arrows
2956,928
Factor Arrows
2478,464
Factor Arrows
2239,232
Factor Arrows
2119,616
Factor Arrows
259,808
Factor Arrows
229,904
Factor Arrows
214,952
Factor Arrows
27,476
Factor Arrows
23,738
Factor Arrows
21,869
Factor Arrows
3623
Factor Arrows
789

More Prime Factorization Examples

3,827,7103,827,7113,827,7133,827,714
21 x 51 x 291 x 671 x 1971831 x 1071 x 4311531 x 72,221121 x 112 x 15,8171

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