Q: What is the prime factorization of the number 213,055,033?

Why is the prime factorization of 213,055,033 written as 17¹ x 31¹ x 523¹ x 773¹?

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 213,055,033

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 213,055,033 by 2

213,055,033 ÷ 2 = 106,527,516.5 - This has a remainder. Let's try another prime number.
213,055,033 ÷ 3 = 71,018,344.3333 - This has a remainder. Let's try another prime number.
213,055,033 ÷ 5 = 42,611,006.6 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
213,055,033 ÷ 17 = 12,532,649 - No remainder! 17 is one of the factors!
12,532,649 ÷ 17 = 737,214.6471 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
12,532,649 ÷ 19 = 659,613.1053 - This has a remainder. 19 is not a factor.
12,532,649 ÷ 23 = 544,897.7826 - This has a remainder. 23 is not a factor.
12,532,649 ÷ 29 = 432,160.3103 - This has a remainder. 29 is not a factor.
12,532,649 ÷ 31 = 404,279 - No remainder! 31 is one of the factors!
404,279 ÷ 31 = 13,041.2581 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
404,279 ÷ 37 = 10,926.4595 - This has a remainder. 37 is not a factor.
404,279 ÷ 41 = 9,860.4634 - This has a remainder. 41 is not a factor.
404,279 ÷ 43 = 9,401.8372 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
404,279 ÷ 523 = 773 - No remainder! 523 is one of the factors!
773 ÷ 523 = 1.478 - There is a remainder. We can't divide by 523 evenly anymore. Let's try the next prime number
773 ÷ 541 = 1.4288 - This has a remainder. 541 is not a factor.
773 ÷ 547 = 1.4132 - This has a remainder. 547 is not a factor.
773 ÷ 557 = 1.3878 - This has a remainder. 557 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
773 ÷ 773 = 1 - No remainder! 773 is one of the factors!

The orange divisor(s) above are the prime factors of the number 213,055,033. If we put all of it together we have the factors 17 x 31 x 523 x 773 = 213,055,033. It can also be written in exponential form as 17¹ x 31¹ x 523¹ x 773¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 213,055,033.

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