Q: What is the prime factorization of the number 213,503,324?

 A:
  • The prime factors are: 2 x 2 x 31 x 179 x 9,619
    • or also written as { 2, 2, 31, 179, 9,619 }
  • Written in exponential form: 22 x 311 x 1791 x 9,6191

Why is the prime factorization of 213,503,324 written as 22 x 311 x 1791 x 9,6191?

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 213,503,324

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 213,503,324 by 2

213,503,324 ÷ 2 = 106,751,662 - No remainder! 2 is one of the factors!
106,751,662 ÷ 2 = 53,375,831 - No remainder! 2 is one of the factors!
53,375,831 ÷ 2 = 26,687,915.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
53,375,831 ÷ 3 = 17,791,943.6667 - This has a remainder. 3 is not a factor.
53,375,831 ÷ 5 = 10,675,166.2 - This has a remainder. 5 is not a factor.
53,375,831 ÷ 7 = 7,625,118.7143 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
53,375,831 ÷ 31 = 1,721,801 - No remainder! 31 is one of the factors!
1,721,801 ÷ 31 = 55,541.9677 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
1,721,801 ÷ 37 = 46,535.1622 - This has a remainder. 37 is not a factor.
1,721,801 ÷ 41 = 41,995.1463 - This has a remainder. 41 is not a factor.
1,721,801 ÷ 43 = 40,041.8837 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,721,801 ÷ 179 = 9,619 - No remainder! 179 is one of the factors!
9,619 ÷ 179 = 53.7374 - There is a remainder. We can't divide by 179 evenly anymore. Let's try the next prime number
9,619 ÷ 181 = 53.1436 - This has a remainder. 181 is not a factor.
9,619 ÷ 191 = 50.3613 - This has a remainder. 191 is not a factor.
9,619 ÷ 193 = 49.8394 - This has a remainder. 193 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
9,619 ÷ 9,619 = 1 - No remainder! 9,619 is one of the factors!

The orange divisor(s) above are the prime factors of the number 213,503,324. If we put all of it together we have the factors 2 x 2 x 31 x 179 x 9,619 = 213,503,324. It can also be written in exponential form as 22 x 311 x 1791 x 9,6191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 213,503,324.

213,503,324
Factor Arrows
2106,751,662
Factor Arrows
253,375,831
Factor Arrows
311,721,801
Factor Arrows
1799,619

More Prime Factorization Examples

213,503,322213,503,323213,503,325213,503,326
21 x 31 x 2231 x 159,5691111 x 171 x 191 x 60,091131 x 52 x 71 x 406,673121 x 106,751,6631

Try the factor calculator

Explore more about the number 213,503,324:


Ask a Question