Q: What is the prime factorization of the number 416,126,223?

Why is the prime factorization of 416,126,223 written as 3² x 3,863¹ x 11,969¹?

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 416,126,223

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 416,126,223 by 2

416,126,223 ÷ 2 = 208,063,111.5 - This has a remainder. Let's try another prime number.
416,126,223 ÷ 3 = 138,708,741 - No remainder! 3 is one of the factors!
138,708,741 ÷ 3 = 46,236,247 - No remainder! 3 is one of the factors!
46,236,247 ÷ 3 = 15,412,082.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
46,236,247 ÷ 5 = 9,247,249.4 - This has a remainder. 5 is not a factor.
46,236,247 ÷ 7 = 6,605,178.1429 - This has a remainder. 7 is not a factor.
46,236,247 ÷ 11 = 4,203,295.1818 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
46,236,247 ÷ 3,863 = 11,969 - No remainder! 3,863 is one of the factors!
11,969 ÷ 3,863 = 3.0984 - There is a remainder. We can't divide by 3863 evenly anymore. Let's try the next prime number
11,969 ÷ 3,877 = 3.0872 - This has a remainder. 3,877 is not a factor.
11,969 ÷ 3,881 = 3.084 - This has a remainder. 3,881 is not a factor.
11,969 ÷ 3,889 = 3.0777 - This has a remainder. 3,889 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,969 ÷ 11,969 = 1 - No remainder! 11,969 is one of the factors!

The orange divisor(s) above are the prime factors of the number 416,126,223. If we put all of it together we have the factors 3 x 3 x 3,863 x 11,969 = 416,126,223. It can also be written in exponential form as 3² x 3,863¹ x 11,969¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 416,126,223.

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