Q: What is the prime factorization of the number 511,340,112?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 877 x 4,049
    • or also written as { 2, 2, 2, 2, 3, 3, 877, 4,049 }
  • Written in exponential form: 24 x 32 x 8771 x 4,0491

Why is the prime factorization of 511,340,112 written as 24 x 32 x 8771 x 4,0491?

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 511,340,112

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 511,340,112 by 2

511,340,112 ÷ 2 = 255,670,056 - No remainder! 2 is one of the factors!
255,670,056 ÷ 2 = 127,835,028 - No remainder! 2 is one of the factors!
127,835,028 ÷ 2 = 63,917,514 - No remainder! 2 is one of the factors!
63,917,514 ÷ 2 = 31,958,757 - No remainder! 2 is one of the factors!
31,958,757 ÷ 2 = 15,979,378.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
31,958,757 ÷ 3 = 10,652,919 - No remainder! 3 is one of the factors!
10,652,919 ÷ 3 = 3,550,973 - No remainder! 3 is one of the factors!
3,550,973 ÷ 3 = 1,183,657.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,550,973 ÷ 5 = 710,194.6 - This has a remainder. 5 is not a factor.
3,550,973 ÷ 7 = 507,281.8571 - This has a remainder. 7 is not a factor.
3,550,973 ÷ 11 = 322,815.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,550,973 ÷ 877 = 4,049 - No remainder! 877 is one of the factors!
4,049 ÷ 877 = 4.6169 - There is a remainder. We can't divide by 877 evenly anymore. Let's try the next prime number
4,049 ÷ 881 = 4.5959 - This has a remainder. 881 is not a factor.
4,049 ÷ 883 = 4.5855 - This has a remainder. 883 is not a factor.
4,049 ÷ 887 = 4.5648 - This has a remainder. 887 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,049 ÷ 4,049 = 1 - No remainder! 4,049 is one of the factors!

The orange divisor(s) above are the prime factors of the number 511,340,112. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 877 x 4,049 = 511,340,112. It can also be written in exponential form as 24 x 32 x 8771 x 4,0491.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 511,340,112.

511,340,112
Factor Arrows
2255,670,056
Factor Arrows
2127,835,028
Factor Arrows
263,917,514
Factor Arrows
231,958,757
Factor Arrows
310,652,919
Factor Arrows
33,550,973
Factor Arrows
8774,049

More Prime Factorization Examples

511,340,110511,340,111511,340,113511,340,114
21 x 51 x 171 x 411 x 73,3631371 x 3891 x 35,5271511,340,113121 x 671 x 1,0131 x 3,7671

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