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 361,531,421 by 2
361,531,421 ÷ 2 = 180,765,710.5 - This has a remainder. Let's try another prime number.
361,531,421 ÷ 3 = 120,510,473.6667 - This has a remainder. Let's try another prime number.
361,531,421 ÷ 5 = 72,306,284.2 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
361,531,421 ÷ 173 = 2,089,777 - No remainder! 173 is one of the factors!
2,089,777 ÷ 173 = 12,079.6358 - There is a remainder. We can't divide by 173 evenly anymore. Let's try the next prime number
2,089,777 ÷ 179 = 11,674.7318 - This has a remainder. 179 is not a factor.
2,089,777 ÷ 181 = 11,545.7293 - This has a remainder. 181 is not a factor.
2,089,777 ÷ 191 = 10,941.2408 - This has a remainder. 191 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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2,089,777 ÷ 233 = 8,969 - No remainder! 233 is one of the factors!
8,969 ÷ 233 = 38.4936 - There is a remainder. We can't divide by 233 evenly anymore. Let's try the next prime number
8,969 ÷ 239 = 37.5272 - This has a remainder. 239 is not a factor.
8,969 ÷ 241 = 37.2158 - This has a remainder. 241 is not a factor.
8,969 ÷ 251 = 35.7331 - This has a remainder. 251 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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8,969 ÷ 8,969 = 1 - No remainder! 8,969 is one of the factors!
The orange divisor(s) above are the prime factors of the number 361,531,421. If we put all of it together we have the factors 173 x 233 x 8,969 = 361,531,421. It can also be written in exponential form as 1731 x 2331 x 8,9691.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 361,531,421.
| 361,531,421 | |||
 | |||
| 173 | 2,089,777 | ||
| |||
| 233 | 8,969 |
| 361,531,419 | 361,531,420 | 361,531,422 | 361,531,423 |
| 31 x 1011 x 1,193,1731 | 22 x 51 x 4311 x 41,9411 | 21 x 32 x 71 x 2,869,2971 | 112 x 2,987,8631 |