Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.
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 414,052,521 by 2
414,052,521 ÷ 2 = 207,026,260.5 - This has a remainder. Let's try another prime number.
414,052,521 ÷ 3 = 138,017,507 - No remainder! 3 is one of the factors!
138,017,507 ÷ 3 = 46,005,835.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
138,017,507 ÷ 5 = 27,603,501.4 - This has a remainder. 5 is not a factor.
138,017,507 ÷ 7 = 19,716,786.7143 - This has a remainder. 7 is not a factor.
138,017,507 ÷ 11 = 12,547,046.0909 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
138,017,507 ÷ 1,823 = 75,709 - No remainder! 1,823 is one of the factors!
75,709 ÷ 1,823 = 41.5299 - There is a remainder. We can't divide by 1823 evenly anymore. Let's try the next prime number
75,709 ÷ 1,831 = 41.3484 - This has a remainder. 1,831 is not a factor.
75,709 ÷ 1,847 = 40.9903 - This has a remainder. 1,847 is not a factor.
75,709 ÷ 1,861 = 40.6819 - This has a remainder. 1,861 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
75,709 ÷ 75,709 = 1 - No remainder! 75,709 is one of the factors!
The orange divisor(s) above are the prime factors of the number 414,052,521. If we put all of it together we have the factors 3 x 1,823 x 75,709 = 414,052,521. It can also be written in exponential form as 31 x 1,8231 x 75,7091.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 414,052,521.
| 414,052,521 | |||
 | |||
| 3 | 138,017,507 | ||
| |||
| 1,823 | 75,709 |
| 414,052,519 | 414,052,520 | 414,052,522 | 414,052,523 |
| 414,052,5191 | 23 x 51 x 71 x 1,478,7591 | 21 x 131 x 191 x 411 x 20,4431 | 311 x 1091 x 1811 x 6771 |