Q: What is the prime factorization of the number 317,767,212?

 A:
  • The prime factors are: 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7 x 46,703
    • or also written as { 2, 2, 3, 3, 3, 3, 3, 7, 46,703 }
  • Written in exponential form: 22 x 35 x 71 x 46,7031

Why is the prime factorization of 317,767,212 written as 22 x 35 x 71 x 46,7031?

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 317,767,212

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 317,767,212 by 2

317,767,212 ÷ 2 = 158,883,606 - No remainder! 2 is one of the factors!
158,883,606 ÷ 2 = 79,441,803 - No remainder! 2 is one of the factors!
79,441,803 ÷ 2 = 39,720,901.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
79,441,803 ÷ 3 = 26,480,601 - No remainder! 3 is one of the factors!
26,480,601 ÷ 3 = 8,826,867 - No remainder! 3 is one of the factors!
8,826,867 ÷ 3 = 2,942,289 - No remainder! 3 is one of the factors!
2,942,289 ÷ 3 = 980,763 - No remainder! 3 is one of the factors!
980,763 ÷ 3 = 326,921 - No remainder! 3 is one of the factors!
326,921 ÷ 3 = 108,973.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
326,921 ÷ 5 = 65,384.2 - This has a remainder. 5 is not a factor.
326,921 ÷ 7 = 46,703 - No remainder! 7 is one of the factors!
46,703 ÷ 7 = 6,671.8571 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
46,703 ÷ 11 = 4,245.7273 - This has a remainder. 11 is not a factor.
46,703 ÷ 13 = 3,592.5385 - This has a remainder. 13 is not a factor.
46,703 ÷ 17 = 2,747.2353 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
46,703 ÷ 46,703 = 1 - No remainder! 46,703 is one of the factors!

The orange divisor(s) above are the prime factors of the number 317,767,212. If we put all of it together we have the factors 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7 x 46,703 = 317,767,212. It can also be written in exponential form as 22 x 35 x 71 x 46,7031.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 317,767,212.

317,767,212
Factor Arrows
2158,883,606
Factor Arrows
279,441,803
Factor Arrows
326,480,601
Factor Arrows
38,826,867
Factor Arrows
32,942,289
Factor Arrows
3980,763
Factor Arrows
3326,921
Factor Arrows
746,703

More Prime Factorization Examples

317,767,210317,767,211317,767,213317,767,214
21 x 51 x 191 x 291 x 1011 x 5711371 x 1,1291 x 7,6071171 x 18,692,189121 x 1,4271 x 111,3411

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