Q: What is the prime factorization of the number 122,205,520?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 5 x 17 x 59 x 1,523
    • or also written as { 2, 2, 2, 2, 5, 17, 59, 1,523 }
  • Written in exponential form: 24 x 51 x 171 x 591 x 1,5231

Why is the prime factorization of 122,205,520 written as 24 x 51 x 171 x 591 x 1,5231?

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 122,205,520

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.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 122,205,520 by 2

122,205,520 ÷ 2 = 61,102,760 - No remainder! 2 is one of the factors!
61,102,760 ÷ 2 = 30,551,380 - No remainder! 2 is one of the factors!
30,551,380 ÷ 2 = 15,275,690 - No remainder! 2 is one of the factors!
15,275,690 ÷ 2 = 7,637,845 - No remainder! 2 is one of the factors!
7,637,845 ÷ 2 = 3,818,922.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
7,637,845 ÷ 3 = 2,545,948.3333 - This has a remainder. 3 is not a factor.
7,637,845 ÷ 5 = 1,527,569 - No remainder! 5 is one of the factors!
1,527,569 ÷ 5 = 305,513.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,527,569 ÷ 7 = 218,224.1429 - This has a remainder. 7 is not a factor.
1,527,569 ÷ 11 = 138,869.9091 - This has a remainder. 11 is not a factor.
1,527,569 ÷ 13 = 117,505.3077 - This has a remainder. 13 is not a factor.
1,527,569 ÷ 17 = 89,857 - No remainder! 17 is one of the factors!
89,857 ÷ 17 = 5,285.7059 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
89,857 ÷ 19 = 4,729.3158 - This has a remainder. 19 is not a factor.
89,857 ÷ 23 = 3,906.8261 - This has a remainder. 23 is not a factor.
89,857 ÷ 29 = 3,098.5172 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
89,857 ÷ 59 = 1,523 - No remainder! 59 is one of the factors!
1,523 ÷ 59 = 25.8136 - There is a remainder. We can't divide by 59 evenly anymore. Let's try the next prime number
1,523 ÷ 61 = 24.9672 - This has a remainder. 61 is not a factor.
1,523 ÷ 67 = 22.7313 - This has a remainder. 67 is not a factor.
1,523 ÷ 71 = 21.4507 - This has a remainder. 71 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,523 ÷ 1,523 = 1 - No remainder! 1,523 is one of the factors!

The orange divisor(s) above are the prime factors of the number 122,205,520. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 5 x 17 x 59 x 1,523 = 122,205,520. It can also be written in exponential form as 24 x 51 x 171 x 591 x 1,5231.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 122,205,520.

122,205,520
Factor Arrows
261,102,760
Factor Arrows
230,551,380
Factor Arrows
215,275,690
Factor Arrows
27,637,845
Factor Arrows
51,527,569
Factor Arrows
1789,857
Factor Arrows
591,523

More Prime Factorization Examples

122,205,518122,205,519122,205,521122,205,522
21 x 3791 x 161,221132 x 4091 x 33,1991671 x 1,823,963121 x 31 x 20,367,5871

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