Q: What is the prime factorization of the number 405,333,132?

Why is the prime factorization of 405,333,132 written as 2² x 3¹ x 137¹ x 373¹ x 661¹?

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 405,333,132

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 405,333,132 by 2

405,333,132 ÷ 2 = 202,666,566 - No remainder! 2 is one of the factors!
202,666,566 ÷ 2 = 101,333,283 - No remainder! 2 is one of the factors!
101,333,283 ÷ 2 = 50,666,641.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
101,333,283 ÷ 3 = 33,777,761 - No remainder! 3 is one of the factors!
33,777,761 ÷ 3 = 11,259,253.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
33,777,761 ÷ 5 = 6,755,552.2 - This has a remainder. 5 is not a factor.
33,777,761 ÷ 7 = 4,825,394.4286 - This has a remainder. 7 is not a factor.
33,777,761 ÷ 11 = 3,070,705.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
33,777,761 ÷ 137 = 246,553 - No remainder! 137 is one of the factors!
246,553 ÷ 137 = 1,799.6569 - There is a remainder. We can't divide by 137 evenly anymore. Let's try the next prime number
246,553 ÷ 139 = 1,773.7626 - This has a remainder. 139 is not a factor.
246,553 ÷ 149 = 1,654.7181 - This has a remainder. 149 is not a factor.
246,553 ÷ 151 = 1,632.8013 - This has a remainder. 151 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
246,553 ÷ 373 = 661 - No remainder! 373 is one of the factors!
661 ÷ 373 = 1.7721 - There is a remainder. We can't divide by 373 evenly anymore. Let's try the next prime number
661 ÷ 379 = 1.7441 - This has a remainder. 379 is not a factor.
661 ÷ 383 = 1.7258 - This has a remainder. 383 is not a factor.
661 ÷ 389 = 1.6992 - This has a remainder. 389 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
661 ÷ 661 = 1 - No remainder! 661 is one of the factors!

The orange divisor(s) above are the prime factors of the number 405,333,132. If we put all of it together we have the factors 2 x 2 x 3 x 137 x 373 x 661 = 405,333,132. It can also be written in exponential form as 2² x 3¹ x 137¹ x 373¹ x 661¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 405,333,132.

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