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 312,024,019 by 2
312,024,019 ÷ 2 = 156,012,009.5 - This has a remainder. Let's try another prime number.
312,024,019 ÷ 3 = 104,008,006.3333 - This has a remainder. Let's try another prime number.
312,024,019 ÷ 5 = 62,404,803.8 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
312,024,019 ÷ 449 = 694,931 - No remainder! 449 is one of the factors!
694,931 ÷ 449 = 1,547.7305 - There is a remainder. We can't divide by 449 evenly anymore. Let's try the next prime number
694,931 ÷ 457 = 1,520.6368 - This has a remainder. 457 is not a factor.
694,931 ÷ 461 = 1,507.4425 - This has a remainder. 461 is not a factor.
694,931 ÷ 463 = 1,500.9309 - This has a remainder. 463 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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694,931 ÷ 809 = 859 - No remainder! 809 is one of the factors!
859 ÷ 809 = 1.0618 - There is a remainder. We can't divide by 809 evenly anymore. Let's try the next prime number
859 ÷ 811 = 1.0592 - This has a remainder. 811 is not a factor.
859 ÷ 821 = 1.0463 - This has a remainder. 821 is not a factor.
859 ÷ 823 = 1.0437 - This has a remainder. 823 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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859 ÷ 859 = 1 - No remainder! 859 is one of the factors!
The orange divisor(s) above are the prime factors of the number 312,024,019. If we put all of it together we have the factors 449 x 809 x 859 = 312,024,019. It can also be written in exponential form as 4491 x 8091 x 8591.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 312,024,019.
| 312,024,019 | |||
 | |||
| 449 | 694,931 | ||
| |||
| 809 | 859 |
| 312,024,017 | 312,024,018 | 312,024,020 | 312,024,021 |
| 312,024,0171 | 21 x 31 x 171 x 3,059,0591 | 22 x 51 x 71 x 111 x 202,6131 | 31 x 291 x 311 x 115,6931 |