Q: What is the prime factorization of the number 660,511,720?

 A:
  • The prime factors are: 2 x 2 x 2 x 5 x 11 x 89 x 101 x 167
    • or also written as { 2, 2, 2, 5, 11, 89, 101, 167 }
  • Written in exponential form: 23 x 51 x 111 x 891 x 1011 x 1671

Why is the prime factorization of 660,511,720 written as 23 x 51 x 111 x 891 x 1011 x 1671?

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 660,511,720

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 660,511,720 by 2

660,511,720 ÷ 2 = 330,255,860 - No remainder! 2 is one of the factors!
330,255,860 ÷ 2 = 165,127,930 - No remainder! 2 is one of the factors!
165,127,930 ÷ 2 = 82,563,965 - No remainder! 2 is one of the factors!
82,563,965 ÷ 2 = 41,281,982.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
82,563,965 ÷ 3 = 27,521,321.6667 - This has a remainder. 3 is not a factor.
82,563,965 ÷ 5 = 16,512,793 - No remainder! 5 is one of the factors!
16,512,793 ÷ 5 = 3,302,558.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
16,512,793 ÷ 7 = 2,358,970.4286 - This has a remainder. 7 is not a factor.
16,512,793 ÷ 11 = 1,501,163 - No remainder! 11 is one of the factors!
1,501,163 ÷ 11 = 136,469.3636 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
1,501,163 ÷ 13 = 115,474.0769 - This has a remainder. 13 is not a factor.
1,501,163 ÷ 17 = 88,303.7059 - This has a remainder. 17 is not a factor.
1,501,163 ÷ 19 = 79,008.5789 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,501,163 ÷ 89 = 16,867 - No remainder! 89 is one of the factors!
16,867 ÷ 89 = 189.5169 - There is a remainder. We can't divide by 89 evenly anymore. Let's try the next prime number
16,867 ÷ 97 = 173.8866 - This has a remainder. 97 is not a factor.
16,867 ÷ 101 = 167 - No remainder! 101 is one of the factors!
167 ÷ 101 = 1.6535 - There is a remainder. We can't divide by 101 evenly anymore. Let's try the next prime number
167 ÷ 103 = 1.6214 - This has a remainder. 103 is not a factor.
167 ÷ 107 = 1.5607 - This has a remainder. 107 is not a factor.
167 ÷ 109 = 1.5321 - This has a remainder. 109 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
167 ÷ 167 = 1 - No remainder! 167 is one of the factors!

The orange divisor(s) above are the prime factors of the number 660,511,720. If we put all of it together we have the factors 2 x 2 x 2 x 5 x 11 x 89 x 101 x 167 = 660,511,720. It can also be written in exponential form as 23 x 51 x 111 x 891 x 1011 x 1671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 660,511,720.

660,511,720
Factor Arrows
2330,255,860
Factor Arrows
2165,127,930
Factor Arrows
282,563,965
Factor Arrows
516,512,793
Factor Arrows
111,501,163
Factor Arrows
8916,867
Factor Arrows
101167

More Prime Factorization Examples

660,511,718660,511,719660,511,721660,511,722
21 x 3,5831 x 92,173133 x 72 x 499,2531611 x 8631 x 12,547121 x 31 x 131 x 411 x 711 x 2,9091

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