Q: What is the prime factorization of the number 513,646,453?

Why is the prime factorization of 513,646,453 written as 601¹ x 757¹ x 1,129¹?

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 513,646,453

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 513,646,453 by 2

513,646,453 ÷ 2 = 256,823,226.5 - This has a remainder. Let's try another prime number.
513,646,453 ÷ 3 = 171,215,484.3333 - This has a remainder. Let's try another prime number.
513,646,453 ÷ 5 = 102,729,290.6 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
513,646,453 ÷ 601 = 854,653 - No remainder! 601 is one of the factors!
854,653 ÷ 601 = 1,422.0516 - There is a remainder. We can't divide by 601 evenly anymore. Let's try the next prime number
854,653 ÷ 607 = 1,407.9951 - This has a remainder. 607 is not a factor.
854,653 ÷ 613 = 1,394.2137 - This has a remainder. 613 is not a factor.
854,653 ÷ 617 = 1,385.175 - This has a remainder. 617 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
854,653 ÷ 757 = 1,129 - No remainder! 757 is one of the factors!
1,129 ÷ 757 = 1.4914 - There is a remainder. We can't divide by 757 evenly anymore. Let's try the next prime number
1,129 ÷ 761 = 1.4836 - This has a remainder. 761 is not a factor.
1,129 ÷ 769 = 1.4681 - This has a remainder. 769 is not a factor.
1,129 ÷ 773 = 1.4605 - This has a remainder. 773 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,129 ÷ 1,129 = 1 - No remainder! 1,129 is one of the factors!

The orange divisor(s) above are the prime factors of the number 513,646,453. If we put all of it together we have the factors 601 x 757 x 1,129 = 513,646,453. It can also be written in exponential form as 601¹ x 757¹ x 1,129¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 513,646,453.

	513,646,453
601		854,653
	757		1,129

More Prime Factorization Examples

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