Q: What is the prime factorization of the number 121,152,512?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 13 x 19 x 479
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 13, 19, 479 }
  • Written in exponential form: 210 x 131 x 191 x 4791

Why is the prime factorization of 121,152,512 written as 210 x 131 x 191 x 4791?

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 121,152,512

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 121,152,512 by 2

121,152,512 ÷ 2 = 60,576,256 - No remainder! 2 is one of the factors!
60,576,256 ÷ 2 = 30,288,128 - No remainder! 2 is one of the factors!
30,288,128 ÷ 2 = 15,144,064 - No remainder! 2 is one of the factors!
15,144,064 ÷ 2 = 7,572,032 - No remainder! 2 is one of the factors!
7,572,032 ÷ 2 = 3,786,016 - No remainder! 2 is one of the factors!
3,786,016 ÷ 2 = 1,893,008 - No remainder! 2 is one of the factors!
1,893,008 ÷ 2 = 946,504 - No remainder! 2 is one of the factors!
946,504 ÷ 2 = 473,252 - No remainder! 2 is one of the factors!
473,252 ÷ 2 = 236,626 - No remainder! 2 is one of the factors!
236,626 ÷ 2 = 118,313 - No remainder! 2 is one of the factors!
118,313 ÷ 2 = 59,156.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
118,313 ÷ 3 = 39,437.6667 - This has a remainder. 3 is not a factor.
118,313 ÷ 5 = 23,662.6 - This has a remainder. 5 is not a factor.
118,313 ÷ 7 = 16,901.8571 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
118,313 ÷ 13 = 9,101 - No remainder! 13 is one of the factors!
9,101 ÷ 13 = 700.0769 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
9,101 ÷ 17 = 535.3529 - This has a remainder. 17 is not a factor.
9,101 ÷ 19 = 479 - No remainder! 19 is one of the factors!
479 ÷ 19 = 25.2105 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
479 ÷ 23 = 20.8261 - This has a remainder. 23 is not a factor.
479 ÷ 29 = 16.5172 - This has a remainder. 29 is not a factor.
479 ÷ 31 = 15.4516 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
479 ÷ 479 = 1 - No remainder! 479 is one of the factors!

The orange divisor(s) above are the prime factors of the number 121,152,512. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 13 x 19 x 479 = 121,152,512. It can also be written in exponential form as 210 x 131 x 191 x 4791.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 121,152,512.

121,152,512
Factor Arrows
260,576,256
Factor Arrows
230,288,128
Factor Arrows
215,144,064
Factor Arrows
27,572,032
Factor Arrows
23,786,016
Factor Arrows
21,893,008
Factor Arrows
2946,504
Factor Arrows
2473,252
Factor Arrows
2236,626
Factor Arrows
2118,313
Factor Arrows
139,101
Factor Arrows
19479

More Prime Factorization Examples

121,152,510121,152,511121,152,513121,152,514
21 x 36 x 51 x 16,6191471 x 3371 x 7,649131 x 5571 x 72,503121 x 71 x 1031 x 84,0171

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