Q: What is the prime factorization of the number 7,852,320?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 7 x 19 x 41
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 5, 7, 19, 41 }
  • Written in exponential form: 25 x 32 x 51 x 71 x 191 x 411

Why is the prime factorization of 7,852,320 written as 25 x 32 x 51 x 71 x 191 x 411?

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 7,852,320

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 7,852,320 by 2

7,852,320 ÷ 2 = 3,926,160 - No remainder! 2 is one of the factors!
3,926,160 ÷ 2 = 1,963,080 - No remainder! 2 is one of the factors!
1,963,080 ÷ 2 = 981,540 - No remainder! 2 is one of the factors!
981,540 ÷ 2 = 490,770 - No remainder! 2 is one of the factors!
490,770 ÷ 2 = 245,385 - No remainder! 2 is one of the factors!
245,385 ÷ 2 = 122,692.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
245,385 ÷ 3 = 81,795 - No remainder! 3 is one of the factors!
81,795 ÷ 3 = 27,265 - No remainder! 3 is one of the factors!
27,265 ÷ 3 = 9,088.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
27,265 ÷ 5 = 5,453 - No remainder! 5 is one of the factors!
5,453 ÷ 5 = 1,090.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
5,453 ÷ 7 = 779 - No remainder! 7 is one of the factors!
779 ÷ 7 = 111.2857 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
779 ÷ 11 = 70.8182 - This has a remainder. 11 is not a factor.
779 ÷ 13 = 59.9231 - This has a remainder. 13 is not a factor.
779 ÷ 17 = 45.8235 - This has a remainder. 17 is not a factor.
779 ÷ 19 = 41 - No remainder! 19 is one of the factors!
41 ÷ 19 = 2.1579 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
41 ÷ 23 = 1.7826 - This has a remainder. 23 is not a factor.
41 ÷ 29 = 1.4138 - This has a remainder. 29 is not a factor.
41 ÷ 31 = 1.3226 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
41 ÷ 41 = 1 - No remainder! 41 is one of the factors!

The orange divisor(s) above are the prime factors of the number 7,852,320. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 7 x 19 x 41 = 7,852,320. It can also be written in exponential form as 25 x 32 x 51 x 71 x 191 x 411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 7,852,320.

7,852,320
Factor Arrows
23,926,160
Factor Arrows
21,963,080
Factor Arrows
2981,540
Factor Arrows
2490,770
Factor Arrows
2245,385
Factor Arrows
381,795
Factor Arrows
327,265
Factor Arrows
55,453
Factor Arrows
7779
Factor Arrows
1941

More Prime Factorization Examples

7,852,3187,852,3197,852,3217,852,322
21 x 731 x 53,78317,852,3191531 x 148,157121 x 5231 x 7,5071

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