Q: What is the prime factorization of the number 85,291,008?

 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 2 x 3 x 11 x 631
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 631 }
  • Written in exponential form: 212 x 31 x 111 x 6311

Why is the prime factorization of 85,291,008 written as 212 x 31 x 111 x 6311?

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 85,291,008

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 85,291,008 by 2

85,291,008 ÷ 2 = 42,645,504 - No remainder! 2 is one of the factors!
42,645,504 ÷ 2 = 21,322,752 - No remainder! 2 is one of the factors!
21,322,752 ÷ 2 = 10,661,376 - No remainder! 2 is one of the factors!
10,661,376 ÷ 2 = 5,330,688 - No remainder! 2 is one of the factors!
5,330,688 ÷ 2 = 2,665,344 - No remainder! 2 is one of the factors!
2,665,344 ÷ 2 = 1,332,672 - No remainder! 2 is one of the factors!
1,332,672 ÷ 2 = 666,336 - No remainder! 2 is one of the factors!
666,336 ÷ 2 = 333,168 - No remainder! 2 is one of the factors!
333,168 ÷ 2 = 166,584 - No remainder! 2 is one of the factors!
166,584 ÷ 2 = 83,292 - No remainder! 2 is one of the factors!
83,292 ÷ 2 = 41,646 - No remainder! 2 is one of the factors!
41,646 ÷ 2 = 20,823 - No remainder! 2 is one of the factors!
20,823 ÷ 2 = 10,411.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
20,823 ÷ 3 = 6,941 - No remainder! 3 is one of the factors!
6,941 ÷ 3 = 2,313.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,941 ÷ 5 = 1,388.2 - This has a remainder. 5 is not a factor.
6,941 ÷ 7 = 991.5714 - This has a remainder. 7 is not a factor.
6,941 ÷ 11 = 631 - No remainder! 11 is one of the factors!
631 ÷ 11 = 57.3636 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
631 ÷ 13 = 48.5385 - This has a remainder. 13 is not a factor.
631 ÷ 17 = 37.1176 - This has a remainder. 17 is not a factor.
631 ÷ 19 = 33.2105 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
631 ÷ 631 = 1 - No remainder! 631 is one of the factors!

The orange divisor(s) above are the prime factors of the number 85,291,008. 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 2 x 3 x 11 x 631 = 85,291,008. It can also be written in exponential form as 212 x 31 x 111 x 6311.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 85,291,008.

85,291,008
Factor Arrows
242,645,504
Factor Arrows
221,322,752
Factor Arrows
210,661,376
Factor Arrows
25,330,688
Factor Arrows
22,665,344
Factor Arrows
21,332,672
Factor Arrows
2666,336
Factor Arrows
2333,168
Factor Arrows
2166,584
Factor Arrows
283,292
Factor Arrows
241,646
Factor Arrows
220,823
Factor Arrows
36,941
Factor Arrows
11631

More Prime Factorization Examples

85,291,00685,291,00785,291,00985,291,010
21 x 171 x 2,508,5591791 x 1,079,633185,291,009121 x 51 x 71 x 731 x 16,6911

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