Q: What is the prime factorization of the number 720,600,831?

 A:
  • The prime factors are: 3 x 3 x 227 x 521 x 677
    • or also written as { 3, 3, 227, 521, 677 }
  • Written in exponential form: 32 x 2271 x 5211 x 6771

Why is the prime factorization of 720,600,831 written as 32 x 2271 x 5211 x 6771?

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 720,600,831

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 720,600,831 by 2

720,600,831 ÷ 2 = 360,300,415.5 - This has a remainder. Let's try another prime number.
720,600,831 ÷ 3 = 240,200,277 - No remainder! 3 is one of the factors!
240,200,277 ÷ 3 = 80,066,759 - No remainder! 3 is one of the factors!
80,066,759 ÷ 3 = 26,688,919.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
80,066,759 ÷ 5 = 16,013,351.8 - This has a remainder. 5 is not a factor.
80,066,759 ÷ 7 = 11,438,108.4286 - This has a remainder. 7 is not a factor.
80,066,759 ÷ 11 = 7,278,796.2727 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
80,066,759 ÷ 227 = 352,717 - No remainder! 227 is one of the factors!
352,717 ÷ 227 = 1,553.8194 - There is a remainder. We can't divide by 227 evenly anymore. Let's try the next prime number
352,717 ÷ 229 = 1,540.2489 - This has a remainder. 229 is not a factor.
352,717 ÷ 233 = 1,513.8069 - This has a remainder. 233 is not a factor.
352,717 ÷ 239 = 1,475.8033 - This has a remainder. 239 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
352,717 ÷ 521 = 677 - No remainder! 521 is one of the factors!
677 ÷ 521 = 1.2994 - There is a remainder. We can't divide by 521 evenly anymore. Let's try the next prime number
677 ÷ 523 = 1.2945 - This has a remainder. 523 is not a factor.
677 ÷ 541 = 1.2514 - This has a remainder. 541 is not a factor.
677 ÷ 547 = 1.2377 - This has a remainder. 547 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
677 ÷ 677 = 1 - No remainder! 677 is one of the factors!

The orange divisor(s) above are the prime factors of the number 720,600,831. If we put all of it together we have the factors 3 x 3 x 227 x 521 x 677 = 720,600,831. It can also be written in exponential form as 32 x 2271 x 5211 x 6771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 720,600,831.

720,600,831
Factor Arrows
3240,200,277
Factor Arrows
380,066,759
Factor Arrows
227352,717
Factor Arrows
521677

More Prime Factorization Examples

720,600,829720,600,830720,600,832720,600,833
131 x 55,430,833121 x 51 x 411 x 3671 x 4,789128 x 71 x 1631 x 2,4671231 x 31,330,4711

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