Q: What is the prime factorization of the number 413,422,309?

Why is the prime factorization of 413,422,309 written as 23¹ x 1,747¹ x 10,289¹?

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 413,422,309

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 413,422,309 by 2

413,422,309 ÷ 2 = 206,711,154.5 - This has a remainder. Let's try another prime number.
413,422,309 ÷ 3 = 137,807,436.3333 - This has a remainder. Let's try another prime number.
413,422,309 ÷ 5 = 82,684,461.8 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
413,422,309 ÷ 23 = 17,974,883 - No remainder! 23 is one of the factors!
17,974,883 ÷ 23 = 781,516.6522 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
17,974,883 ÷ 29 = 619,823.5517 - This has a remainder. 29 is not a factor.
17,974,883 ÷ 31 = 579,834.9355 - This has a remainder. 31 is not a factor.
17,974,883 ÷ 37 = 485,807.6486 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
17,974,883 ÷ 1,747 = 10,289 - No remainder! 1,747 is one of the factors!
10,289 ÷ 1,747 = 5.8895 - There is a remainder. We can't divide by 1747 evenly anymore. Let's try the next prime number
10,289 ÷ 1,753 = 5.8694 - This has a remainder. 1,753 is not a factor.
10,289 ÷ 1,759 = 5.8493 - This has a remainder. 1,759 is not a factor.
10,289 ÷ 1,777 = 5.7901 - This has a remainder. 1,777 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
10,289 ÷ 10,289 = 1 - No remainder! 10,289 is one of the factors!

The orange divisor(s) above are the prime factors of the number 413,422,309. If we put all of it together we have the factors 23 x 1,747 x 10,289 = 413,422,309. It can also be written in exponential form as 23¹ x 1,747¹ x 10,289¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 413,422,309.

	413,422,309
23		17,974,883
	1,747		10,289

More Prime Factorization Examples

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