Q: What is the prime factorization of the number 312,616,503?

Why is the prime factorization of 312,616,503 written as 3⁴ x 181¹ x 21,323¹?

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 312,616,503

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 312,616,503 by 2

312,616,503 ÷ 2 = 156,308,251.5 - This has a remainder. Let's try another prime number.
312,616,503 ÷ 3 = 104,205,501 - No remainder! 3 is one of the factors!
104,205,501 ÷ 3 = 34,735,167 - No remainder! 3 is one of the factors!
34,735,167 ÷ 3 = 11,578,389 - No remainder! 3 is one of the factors!
11,578,389 ÷ 3 = 3,859,463 - No remainder! 3 is one of the factors!
3,859,463 ÷ 3 = 1,286,487.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
3,859,463 ÷ 5 = 771,892.6 - This has a remainder. 5 is not a factor.
3,859,463 ÷ 7 = 551,351.8571 - This has a remainder. 7 is not a factor.
3,859,463 ÷ 11 = 350,860.2727 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
3,859,463 ÷ 181 = 21,323 - No remainder! 181 is one of the factors!
21,323 ÷ 181 = 117.8066 - There is a remainder. We can't divide by 181 evenly anymore. Let's try the next prime number
21,323 ÷ 191 = 111.6387 - This has a remainder. 191 is not a factor.
21,323 ÷ 193 = 110.4819 - This has a remainder. 193 is not a factor.
21,323 ÷ 197 = 108.2386 - This has a remainder. 197 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
21,323 ÷ 21,323 = 1 - No remainder! 21,323 is one of the factors!

The orange divisor(s) above are the prime factors of the number 312,616,503. If we put all of it together we have the factors 3 x 3 x 3 x 3 x 181 x 21,323 = 312,616,503. It can also be written in exponential form as 3⁴ x 181¹ x 21,323¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 312,616,503.

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