Q: What is the prime factorization of the number 745,475,669?

Why is the prime factorization of 745,475,669 written as 20,117¹ x 37,057¹?

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 745,475,669

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 745,475,669 by 2

745,475,669 ÷ 2 = 372,737,834.5 - This has a remainder. Let's try another prime number.
745,475,669 ÷ 3 = 248,491,889.6667 - This has a remainder. Let's try another prime number.
745,475,669 ÷ 5 = 149,095,133.8 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
745,475,669 ÷ 20,117 = 37,057 - No remainder! 20,117 is one of the factors!
37,057 ÷ 20,117 = 1.8421 - There is a remainder. We can't divide by 20117 evenly anymore. Let's try the next prime number
37,057 ÷ 20,123 = 1.8415 - This has a remainder. 20,123 is not a factor.
37,057 ÷ 20,129 = 1.841 - This has a remainder. 20,129 is not a factor.
37,057 ÷ 20,143 = 1.8397 - This has a remainder. 20,143 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
37,057 ÷ 37,057 = 1 - No remainder! 37,057 is one of the factors!

The orange divisor(s) above are the prime factors of the number 745,475,669. If we put all of it together we have the factors 20,117 x 37,057 = 745,475,669. It can also be written in exponential form as 20,117¹ x 37,057¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 745,475,669.

	745,475,669
20,117		37,057

More Prime Factorization Examples

Try the factor calculator