Q: What is the prime factorization of the number 687,652,416?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 11 x 31 x 389
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 11, 31, 389 }
  • Written in exponential form: 26 x 34 x 111 x 311 x 3891

Why is the prime factorization of 687,652,416 written as 26 x 34 x 111 x 311 x 3891?

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 687,652,416

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 687,652,416 by 2

687,652,416 ÷ 2 = 343,826,208 - No remainder! 2 is one of the factors!
343,826,208 ÷ 2 = 171,913,104 - No remainder! 2 is one of the factors!
171,913,104 ÷ 2 = 85,956,552 - No remainder! 2 is one of the factors!
85,956,552 ÷ 2 = 42,978,276 - No remainder! 2 is one of the factors!
42,978,276 ÷ 2 = 21,489,138 - No remainder! 2 is one of the factors!
21,489,138 ÷ 2 = 10,744,569 - No remainder! 2 is one of the factors!
10,744,569 ÷ 2 = 5,372,284.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,744,569 ÷ 3 = 3,581,523 - No remainder! 3 is one of the factors!
3,581,523 ÷ 3 = 1,193,841 - No remainder! 3 is one of the factors!
1,193,841 ÷ 3 = 397,947 - No remainder! 3 is one of the factors!
397,947 ÷ 3 = 132,649 - No remainder! 3 is one of the factors!
132,649 ÷ 3 = 44,216.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
132,649 ÷ 5 = 26,529.8 - This has a remainder. 5 is not a factor.
132,649 ÷ 7 = 18,949.8571 - This has a remainder. 7 is not a factor.
132,649 ÷ 11 = 12,059 - No remainder! 11 is one of the factors!
12,059 ÷ 11 = 1,096.2727 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
12,059 ÷ 13 = 927.6154 - This has a remainder. 13 is not a factor.
12,059 ÷ 17 = 709.3529 - This has a remainder. 17 is not a factor.
12,059 ÷ 19 = 634.6842 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
12,059 ÷ 31 = 389 - No remainder! 31 is one of the factors!
389 ÷ 31 = 12.5484 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
389 ÷ 37 = 10.5135 - This has a remainder. 37 is not a factor.
389 ÷ 41 = 9.4878 - This has a remainder. 41 is not a factor.
389 ÷ 43 = 9.0465 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
389 ÷ 389 = 1 - No remainder! 389 is one of the factors!

The orange divisor(s) above are the prime factors of the number 687,652,416. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 11 x 31 x 389 = 687,652,416. It can also be written in exponential form as 26 x 34 x 111 x 311 x 3891.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 687,652,416.

687,652,416
Factor Arrows
2343,826,208
Factor Arrows
2171,913,104
Factor Arrows
285,956,552
Factor Arrows
242,978,276
Factor Arrows
221,489,138
Factor Arrows
210,744,569
Factor Arrows
33,581,523
Factor Arrows
31,193,841
Factor Arrows
3397,947
Factor Arrows
3132,649
Factor Arrows
1112,059
Factor Arrows
31389

More Prime Factorization Examples

687,652,414687,652,415687,652,417687,652,418
21 x 171 x 20,225,071151 x 137,530,48311,8011 x 381,817121 x 343,826,2091

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