Q: What is the prime factorization of the number 687,071,232?

 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 2 x 3 x 3 x 9,319
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 9,319 }
  • Written in exponential form: 213 x 32 x 9,3191

Why is the prime factorization of 687,071,232 written as 213 x 32 x 9,3191?

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 687,071,232

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 687,071,232 by 2

687,071,232 ÷ 2 = 343,535,616 - No remainder! 2 is one of the factors!
343,535,616 ÷ 2 = 171,767,808 - No remainder! 2 is one of the factors!
171,767,808 ÷ 2 = 85,883,904 - No remainder! 2 is one of the factors!
85,883,904 ÷ 2 = 42,941,952 - No remainder! 2 is one of the factors!
42,941,952 ÷ 2 = 21,470,976 - No remainder! 2 is one of the factors!
21,470,976 ÷ 2 = 10,735,488 - No remainder! 2 is one of the factors!
10,735,488 ÷ 2 = 5,367,744 - No remainder! 2 is one of the factors!
5,367,744 ÷ 2 = 2,683,872 - No remainder! 2 is one of the factors!
2,683,872 ÷ 2 = 1,341,936 - No remainder! 2 is one of the factors!
1,341,936 ÷ 2 = 670,968 - No remainder! 2 is one of the factors!
670,968 ÷ 2 = 335,484 - No remainder! 2 is one of the factors!
335,484 ÷ 2 = 167,742 - No remainder! 2 is one of the factors!
167,742 ÷ 2 = 83,871 - No remainder! 2 is one of the factors!
83,871 ÷ 2 = 41,935.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
83,871 ÷ 3 = 27,957 - No remainder! 3 is one of the factors!
27,957 ÷ 3 = 9,319 - No remainder! 3 is one of the factors!
9,319 ÷ 3 = 3,106.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
9,319 ÷ 5 = 1,863.8 - This has a remainder. 5 is not a factor.
9,319 ÷ 7 = 1,331.2857 - This has a remainder. 7 is not a factor.
9,319 ÷ 11 = 847.1818 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
9,319 ÷ 9,319 = 1 - No remainder! 9,319 is one of the factors!

The orange divisor(s) above are the prime factors of the number 687,071,232. 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 2 x 3 x 3 x 9,319 = 687,071,232. It can also be written in exponential form as 213 x 32 x 9,3191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 687,071,232.

687,071,232
Factor Arrows
2343,535,616
Factor Arrows
2171,767,808
Factor Arrows
285,883,904
Factor Arrows
242,941,952
Factor Arrows
221,470,976
Factor Arrows
210,735,488
Factor Arrows
25,367,744
Factor Arrows
22,683,872
Factor Arrows
21,341,936
Factor Arrows
2670,968
Factor Arrows
2335,484
Factor Arrows
2167,742
Factor Arrows
283,871
Factor Arrows
327,957
Factor Arrows
39,319

More Prime Factorization Examples

687,071,230687,071,231687,071,233687,071,234
21 x 51 x 1731 x 397,151171 x 111 x 8,923,00311671 x 4,114,199121 x 6411 x 535,9371

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