Q: What is the prime factorization of the number 400,414,219?

Why is the prime factorization of 400,414,219 written as 811¹ x 493,729¹?

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 400,414,219

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 400,414,219 by 2

400,414,219 ÷ 2 = 200,207,109.5 - This has a remainder. Let's try another prime number.
400,414,219 ÷ 3 = 133,471,406.3333 - This has a remainder. Let's try another prime number.
400,414,219 ÷ 5 = 80,082,843.8 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
400,414,219 ÷ 811 = 493,729 - No remainder! 811 is one of the factors!
493,729 ÷ 811 = 608.7904 - There is a remainder. We can't divide by 811 evenly anymore. Let's try the next prime number
493,729 ÷ 821 = 601.3752 - This has a remainder. 821 is not a factor.
493,729 ÷ 823 = 599.9137 - This has a remainder. 823 is not a factor.
493,729 ÷ 827 = 597.0121 - This has a remainder. 827 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
493,729 ÷ 493,729 = 1 - No remainder! 493,729 is one of the factors!

The orange divisor(s) above are the prime factors of the number 400,414,219. If we put all of it together we have the factors 811 x 493,729 = 400,414,219. It can also be written in exponential form as 811¹ x 493,729¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 400,414,219.

	400,414,219
811		493,729

More Prime Factorization Examples

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