Q: What is the prime factorization of the number 27,614,720?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 7 x 23 x 67
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 7, 23, 67 }
  • Written in exponential form: 29 x 51 x 71 x 231 x 671

Why is the prime factorization of 27,614,720 written as 29 x 51 x 71 x 231 x 671?

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 27,614,720

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 27,614,720 by 2

27,614,720 ÷ 2 = 13,807,360 - No remainder! 2 is one of the factors!
13,807,360 ÷ 2 = 6,903,680 - No remainder! 2 is one of the factors!
6,903,680 ÷ 2 = 3,451,840 - No remainder! 2 is one of the factors!
3,451,840 ÷ 2 = 1,725,920 - No remainder! 2 is one of the factors!
1,725,920 ÷ 2 = 862,960 - No remainder! 2 is one of the factors!
862,960 ÷ 2 = 431,480 - No remainder! 2 is one of the factors!
431,480 ÷ 2 = 215,740 - No remainder! 2 is one of the factors!
215,740 ÷ 2 = 107,870 - No remainder! 2 is one of the factors!
107,870 ÷ 2 = 53,935 - No remainder! 2 is one of the factors!
53,935 ÷ 2 = 26,967.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
53,935 ÷ 3 = 17,978.3333 - This has a remainder. 3 is not a factor.
53,935 ÷ 5 = 10,787 - No remainder! 5 is one of the factors!
10,787 ÷ 5 = 2,157.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
10,787 ÷ 7 = 1,541 - No remainder! 7 is one of the factors!
1,541 ÷ 7 = 220.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
1,541 ÷ 11 = 140.0909 - This has a remainder. 11 is not a factor.
1,541 ÷ 13 = 118.5385 - This has a remainder. 13 is not a factor.
1,541 ÷ 17 = 90.6471 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,541 ÷ 23 = 67 - No remainder! 23 is one of the factors!
67 ÷ 23 = 2.913 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
67 ÷ 29 = 2.3103 - This has a remainder. 29 is not a factor.
67 ÷ 31 = 2.1613 - This has a remainder. 31 is not a factor.
67 ÷ 37 = 1.8108 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
67 ÷ 67 = 1 - No remainder! 67 is one of the factors!

The orange divisor(s) above are the prime factors of the number 27,614,720. 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 5 x 7 x 23 x 67 = 27,614,720. It can also be written in exponential form as 29 x 51 x 71 x 231 x 671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 27,614,720.

27,614,720
Factor Arrows
213,807,360
Factor Arrows
26,903,680
Factor Arrows
23,451,840
Factor Arrows
21,725,920
Factor Arrows
2862,960
Factor Arrows
2431,480
Factor Arrows
2215,740
Factor Arrows
2107,870
Factor Arrows
253,935
Factor Arrows
510,787
Factor Arrows
71,541
Factor Arrows
2367

More Prime Factorization Examples

27,614,71827,614,71927,614,72127,614,722
21 x 32 x 1,534,1511111 x 2,510,429131 x 3891 x 23,663121 x 13,807,3611

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