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 31,132,321 by 2
31,132,321 ÷ 2 = 15,566,160.5 - This has a remainder. Let's try another prime number.
31,132,321 ÷ 3 = 10,377,440.3333 - This has a remainder. Let's try another prime number.
31,132,321 ÷ 5 = 6,226,464.2 - This has a remainder. Let's try another prime number.
31,132,321 ÷ 11 = 2,830,211 - No remainder! 11 is one of the factors!
2,830,211 ÷ 11 = 257,291.9091 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
2,830,211 ÷ 13 = 217,708.5385 - This has a remainder. 13 is not a factor.
2,830,211 ÷ 17 = 166,483 - No remainder! 17 is one of the factors!
166,483 ÷ 17 = 9,793.1176 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
166,483 ÷ 19 = 8,762.2632 - This has a remainder. 19 is not a factor.
166,483 ÷ 23 = 7,238.3913 - This has a remainder. 23 is not a factor.
166,483 ÷ 29 = 5,740.7931 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
166,483 ÷ 229 = 727 - No remainder! 229 is one of the factors!
727 ÷ 229 = 3.1747 - There is a remainder. We can't divide by 229 evenly anymore. Let's try the next prime number
727 ÷ 233 = 3.1202 - This has a remainder. 233 is not a factor.
727 ÷ 239 = 3.0418 - This has a remainder. 239 is not a factor.
727 ÷ 241 = 3.0166 - This has a remainder. 241 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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727 ÷ 727 = 1 - No remainder! 727 is one of the factors!
The orange divisor(s) above are the prime factors of the number 31,132,321. If we put all of it together we have the factors 11 x 17 x 229 x 727 = 31,132,321. It can also be written in exponential form as 111 x 171 x 2291 x 7271.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 31,132,321.
| 31,132,321 | ||||
 | ||||
| 11 | 2,830,211 | |||
| ||||
| 17 | 166,483 | |||
| ||||
| 229 | 727 |
| 31,132,319 | 31,132,320 | 31,132,322 | 31,132,323 |
| 4,5831 x 6,7931 | 25 x 31 x 51 x 791 x 8211 | 21 x 131 x 9971 x 1,2011 | 33 x 1,153,0491 |