Q: What is the prime factorization of the number 313,705,130?

 A:
  • The prime factors are: 2 x 5 x 2,089 x 15,017
    • or also written as { 2, 5, 2,089, 15,017 }
  • Written in exponential form: 21 x 51 x 2,0891 x 15,0171

Why is the prime factorization of 313,705,130 written as 21 x 51 x 2,0891 x 15,0171?

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 313,705,130

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 313,705,130 by 2

313,705,130 ÷ 2 = 156,852,565 - No remainder! 2 is one of the factors!
156,852,565 ÷ 2 = 78,426,282.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
156,852,565 ÷ 3 = 52,284,188.3333 - This has a remainder. 3 is not a factor.
156,852,565 ÷ 5 = 31,370,513 - No remainder! 5 is one of the factors!
31,370,513 ÷ 5 = 6,274,102.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
31,370,513 ÷ 7 = 4,481,501.8571 - This has a remainder. 7 is not a factor.
31,370,513 ÷ 11 = 2,851,864.8182 - This has a remainder. 11 is not a factor.
31,370,513 ÷ 13 = 2,413,116.3846 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
31,370,513 ÷ 2,089 = 15,017 - No remainder! 2,089 is one of the factors!
15,017 ÷ 2,089 = 7.1886 - There is a remainder. We can't divide by 2089 evenly anymore. Let's try the next prime number
15,017 ÷ 2,099 = 7.1544 - This has a remainder. 2,099 is not a factor.
15,017 ÷ 2,111 = 7.1137 - This has a remainder. 2,111 is not a factor.
15,017 ÷ 2,113 = 7.107 - This has a remainder. 2,113 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
15,017 ÷ 15,017 = 1 - No remainder! 15,017 is one of the factors!

The orange divisor(s) above are the prime factors of the number 313,705,130. If we put all of it together we have the factors 2 x 5 x 2,089 x 15,017 = 313,705,130. It can also be written in exponential form as 21 x 51 x 2,0891 x 15,0171.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 313,705,130.

313,705,130
Factor Arrows
2156,852,565
Factor Arrows
531,370,513
Factor Arrows
2,08915,017

More Prime Factorization Examples

313,705,128313,705,129313,705,131313,705,132
23 x 31 x 111 x 1,188,2771371 x 791 x 107,323131 x 171 x 6,151,081122 x 131 x 6,032,7911

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