Q: What is the prime factorization of the number 2,160,952?

 A:
  • The prime factors are: 2 x 2 x 2 x 313 x 863
    • or also written as { 2, 2, 2, 313, 863 }
  • Written in exponential form: 23 x 3131 x 8631

Why is the prime factorization of 2,160,952 written as 23 x 3131 x 8631?

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 2,160,952

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 2,160,952 by 2

2,160,952 ÷ 2 = 1,080,476 - No remainder! 2 is one of the factors!
1,080,476 ÷ 2 = 540,238 - No remainder! 2 is one of the factors!
540,238 ÷ 2 = 270,119 - No remainder! 2 is one of the factors!
270,119 ÷ 2 = 135,059.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
270,119 ÷ 3 = 90,039.6667 - This has a remainder. 3 is not a factor.
270,119 ÷ 5 = 54,023.8 - This has a remainder. 5 is not a factor.
270,119 ÷ 7 = 38,588.4286 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
270,119 ÷ 313 = 863 - No remainder! 313 is one of the factors!
863 ÷ 313 = 2.7572 - There is a remainder. We can't divide by 313 evenly anymore. Let's try the next prime number
863 ÷ 317 = 2.7224 - This has a remainder. 317 is not a factor.
863 ÷ 331 = 2.6073 - This has a remainder. 331 is not a factor.
863 ÷ 337 = 2.5608 - This has a remainder. 337 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
863 ÷ 863 = 1 - No remainder! 863 is one of the factors!

The orange divisor(s) above are the prime factors of the number 2,160,952. If we put all of it together we have the factors 2 x 2 x 2 x 313 x 863 = 2,160,952. It can also be written in exponential form as 23 x 3131 x 8631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 2,160,952.

2,160,952
Factor Arrows
21,080,476
Factor Arrows
2540,238
Factor Arrows
2270,119
Factor Arrows
313863

More Prime Factorization Examples

2,160,9502,160,9512,160,9532,160,954
21 x 52 x 111 x 3,929131 x 131 x 671 x 82712,160,953121 x 32 x 2711 x 4431

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