Q: What is the prime factorization of the number 320,012,123?

 A:
  • The prime factors are: 71 x 439 x 10,267
    • or also written as { 71, 439, 10,267 }
  • Written in exponential form: 711 x 4391 x 10,2671

Why is the prime factorization of 320,012,123 written as 711 x 4391 x 10,2671?

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 320,012,123

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 320,012,123 by 2

320,012,123 ÷ 2 = 160,006,061.5 - This has a remainder. Let's try another prime number.
320,012,123 ÷ 3 = 106,670,707.6667 - This has a remainder. Let's try another prime number.
320,012,123 ÷ 5 = 64,002,424.6 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
320,012,123 ÷ 71 = 4,507,213 - No remainder! 71 is one of the factors!
4,507,213 ÷ 71 = 63,481.8732 - There is a remainder. We can't divide by 71 evenly anymore. Let's try the next prime number
4,507,213 ÷ 73 = 61,742.6438 - This has a remainder. 73 is not a factor.
4,507,213 ÷ 79 = 57,053.3291 - This has a remainder. 79 is not a factor.
4,507,213 ÷ 83 = 54,303.7711 - This has a remainder. 83 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,507,213 ÷ 439 = 10,267 - No remainder! 439 is one of the factors!
10,267 ÷ 439 = 23.3872 - There is a remainder. We can't divide by 439 evenly anymore. Let's try the next prime number
10,267 ÷ 443 = 23.1761 - This has a remainder. 443 is not a factor.
10,267 ÷ 449 = 22.8664 - This has a remainder. 449 is not a factor.
10,267 ÷ 457 = 22.4661 - This has a remainder. 457 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
10,267 ÷ 10,267 = 1 - No remainder! 10,267 is one of the factors!

The orange divisor(s) above are the prime factors of the number 320,012,123. If we put all of it together we have the factors 71 x 439 x 10,267 = 320,012,123. It can also be written in exponential form as 711 x 4391 x 10,2671.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 320,012,123.

320,012,123
Factor Arrows
714,507,213
Factor Arrows
43910,267

More Prime Factorization Examples

320,012,121320,012,122320,012,124320,012,125
31 x 111 x 131 x 5031 x 1,483121 x 5,2371 x 30,553122 x 31 x 3311 x 80,567153 x 4431 x 5,7791

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