Q: What is the prime factorization of the number 412,146,105?

Why is the prime factorization of 412,146,105 written as 3¹ x 5¹ x 7² x 683¹ x 821¹?

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 412,146,105

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 412,146,105 by 2

412,146,105 ÷ 2 = 206,073,052.5 - This has a remainder. Let's try another prime number.
412,146,105 ÷ 3 = 137,382,035 - No remainder! 3 is one of the factors!
137,382,035 ÷ 3 = 45,794,011.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
137,382,035 ÷ 5 = 27,476,407 - No remainder! 5 is one of the factors!
27,476,407 ÷ 5 = 5,495,281.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
27,476,407 ÷ 7 = 3,925,201 - No remainder! 7 is one of the factors!
3,925,201 ÷ 7 = 560,743 - No remainder! 7 is one of the factors!
560,743 ÷ 7 = 80,106.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
560,743 ÷ 11 = 50,976.6364 - This has a remainder. 11 is not a factor.
560,743 ÷ 13 = 43,134.0769 - This has a remainder. 13 is not a factor.
560,743 ÷ 17 = 32,984.8824 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
560,743 ÷ 683 = 821 - No remainder! 683 is one of the factors!
821 ÷ 683 = 1.202 - There is a remainder. We can't divide by 683 evenly anymore. Let's try the next prime number
821 ÷ 691 = 1.1881 - This has a remainder. 691 is not a factor.
821 ÷ 701 = 1.1712 - This has a remainder. 701 is not a factor.
821 ÷ 709 = 1.158 - This has a remainder. 709 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
821 ÷ 821 = 1 - No remainder! 821 is one of the factors!

The orange divisor(s) above are the prime factors of the number 412,146,105. If we put all of it together we have the factors 3 x 5 x 7 x 7 x 683 x 821 = 412,146,105. It can also be written in exponential form as 3¹ x 5¹ x 7² x 683¹ x 821¹.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 412,146,105.

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