Q: What is the prime factorization of the number 156,321,331?

 A:
  • The prime factors are: 233 x 677 x 991
    • or also written as { 233, 677, 991 }
  • Written in exponential form: 2331 x 6771 x 9911

Why is the prime factorization of 156,321,331 written as 2331 x 6771 x 9911?

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 156,321,331

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 156,321,331 by 2

156,321,331 ÷ 2 = 78,160,665.5 - This has a remainder. Let's try another prime number.
156,321,331 ÷ 3 = 52,107,110.3333 - This has a remainder. Let's try another prime number.
156,321,331 ÷ 5 = 31,264,266.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
156,321,331 ÷ 233 = 670,907 - No remainder! 233 is one of the factors!
670,907 ÷ 233 = 2,879.4292 - There is a remainder. We can't divide by 233 evenly anymore. Let's try the next prime number
670,907 ÷ 239 = 2,807.1423 - This has a remainder. 239 is not a factor.
670,907 ÷ 241 = 2,783.8465 - This has a remainder. 241 is not a factor.
670,907 ÷ 251 = 2,672.9363 - This has a remainder. 251 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
670,907 ÷ 677 = 991 - No remainder! 677 is one of the factors!
991 ÷ 677 = 1.4638 - There is a remainder. We can't divide by 677 evenly anymore. Let's try the next prime number
991 ÷ 683 = 1.451 - This has a remainder. 683 is not a factor.
991 ÷ 691 = 1.4342 - This has a remainder. 691 is not a factor.
991 ÷ 701 = 1.4137 - This has a remainder. 701 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
991 ÷ 991 = 1 - No remainder! 991 is one of the factors!

The orange divisor(s) above are the prime factors of the number 156,321,331. If we put all of it together we have the factors 233 x 677 x 991 = 156,321,331. It can also be written in exponential form as 2331 x 6771 x 9911.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 156,321,331.

156,321,331
Factor Arrows
233670,907
Factor Arrows
677991

More Prime Factorization Examples

156,321,329156,321,330156,321,332156,321,333
791 x 1071 x 18,493121 x 31 x 51 x 111 x 3071 x 1,543122 x 8871 x 44,059134 x 71 x 275,6991

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