Q: What is the prime factorization of the number 765,564,781?

Why is the prime factorization of 765,564,781 written as 25,247¹ x 30,323¹?

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 765,564,781

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 765,564,781 by 2

765,564,781 ÷ 2 = 382,782,390.5 - This has a remainder. Let's try another prime number.
765,564,781 ÷ 3 = 255,188,260.3333 - This has a remainder. Let's try another prime number.
765,564,781 ÷ 5 = 153,112,956.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
765,564,781 ÷ 25,247 = 30,323 - No remainder! 25,247 is one of the factors!
30,323 ÷ 25,247 = 1.2011 - There is a remainder. We can't divide by 25247 evenly anymore. Let's try the next prime number
30,323 ÷ 25,253 = 1.2008 - This has a remainder. 25,253 is not a factor.
30,323 ÷ 25,261 = 1.2004 - This has a remainder. 25,261 is not a factor.
30,323 ÷ 25,301 = 1.1985 - This has a remainder. 25,301 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
30,323 ÷ 30,323 = 1 - No remainder! 30,323 is one of the factors!

The orange divisor(s) above are the prime factors of the number 765,564,781. If we put all of it together we have the factors 25,247 x 30,323 = 765,564,781. It can also be written in exponential form as 25,247¹ x 30,323¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 765,564,781.

	765,564,781
25,247		30,323

More Prime Factorization Examples

Try the factor calculator