Q: What is the prime factorization of the number 220,211,123?

Why is the prime factorization of 220,211,123 written as 11¹ x 29¹ x 41¹ x 113¹ x 149¹?

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 220,211,123

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 220,211,123 by 2

220,211,123 ÷ 2 = 110,105,561.5 - This has a remainder. Let's try another prime number.
220,211,123 ÷ 3 = 73,403,707.6667 - This has a remainder. Let's try another prime number.
220,211,123 ÷ 5 = 44,042,224.6 - This has a remainder. Let's try another prime number.
220,211,123 ÷ 11 = 20,019,193 - No remainder! 11 is one of the factors!
20,019,193 ÷ 11 = 1,819,926.6364 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
20,019,193 ÷ 13 = 1,539,937.9231 - This has a remainder. 13 is not a factor.
20,019,193 ÷ 17 = 1,177,599.5882 - This has a remainder. 17 is not a factor.
20,019,193 ÷ 19 = 1,053,641.7368 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
20,019,193 ÷ 29 = 690,317 - No remainder! 29 is one of the factors!
690,317 ÷ 29 = 23,804.0345 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
690,317 ÷ 31 = 22,268.2903 - This has a remainder. 31 is not a factor.
690,317 ÷ 37 = 18,657.2162 - This has a remainder. 37 is not a factor.
690,317 ÷ 41 = 16,837 - No remainder! 41 is one of the factors!
16,837 ÷ 41 = 410.6585 - There is a remainder. We can't divide by 41 evenly anymore. Let's try the next prime number
16,837 ÷ 43 = 391.5581 - This has a remainder. 43 is not a factor.
16,837 ÷ 47 = 358.234 - This has a remainder. 47 is not a factor.
16,837 ÷ 53 = 317.6792 - This has a remainder. 53 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
16,837 ÷ 113 = 149 - No remainder! 113 is one of the factors!
149 ÷ 113 = 1.3186 - There is a remainder. We can't divide by 113 evenly anymore. Let's try the next prime number
149 ÷ 127 = 1.1732 - This has a remainder. 127 is not a factor.
149 ÷ 131 = 1.1374 - This has a remainder. 131 is not a factor.
149 ÷ 137 = 1.0876 - This has a remainder. 137 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
149 ÷ 149 = 1 - No remainder! 149 is one of the factors!

The orange divisor(s) above are the prime factors of the number 220,211,123. If we put all of it together we have the factors 11 x 29 x 41 x 113 x 149 = 220,211,123. It can also be written in exponential form as 11¹ x 29¹ x 41¹ x 113¹ x 149¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 220,211,123.

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