Q: What is the prime factorization of the number 210,253,321?

Why is the prime factorization of 210,253,321 written as 10,303¹ x 20,407¹?

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 210,253,321

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.

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 210,253,321 by 2

210,253,321 ÷ 2 = 105,126,660.5 - This has a remainder. Let's try another prime number.
210,253,321 ÷ 3 = 70,084,440.3333 - This has a remainder. Let's try another prime number.
210,253,321 ÷ 5 = 42,050,664.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
210,253,321 ÷ 10,303 = 20,407 - No remainder! 10,303 is one of the factors!
20,407 ÷ 10,303 = 1.9807 - There is a remainder. We can't divide by 10303 evenly anymore. Let's try the next prime number
20,407 ÷ 10,313 = 1.9788 - This has a remainder. 10,313 is not a factor.
20,407 ÷ 10,321 = 1.9772 - This has a remainder. 10,321 is not a factor.
20,407 ÷ 10,331 = 1.9753 - This has a remainder. 10,331 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
20,407 ÷ 20,407 = 1 - No remainder! 20,407 is one of the factors!

The orange divisor(s) above are the prime factors of the number 210,253,321. If we put all of it together we have the factors 10,303 x 20,407 = 210,253,321. It can also be written in exponential form as 10,303¹ x 20,407¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 210,253,321.

	210,253,321
10,303		20,407

More Prime Factorization Examples

Try the factor calculator