Q: What is the prime factorization of the number 129,704,576?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 233 x 4,349
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 233, 4,349 }
  • Written in exponential form: 27 x 2331 x 4,3491

Why is the prime factorization of 129,704,576 written as 27 x 2331 x 4,3491?

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 129,704,576

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 129,704,576 by 2

129,704,576 ÷ 2 = 64,852,288 - No remainder! 2 is one of the factors!
64,852,288 ÷ 2 = 32,426,144 - No remainder! 2 is one of the factors!
32,426,144 ÷ 2 = 16,213,072 - No remainder! 2 is one of the factors!
16,213,072 ÷ 2 = 8,106,536 - No remainder! 2 is one of the factors!
8,106,536 ÷ 2 = 4,053,268 - No remainder! 2 is one of the factors!
4,053,268 ÷ 2 = 2,026,634 - No remainder! 2 is one of the factors!
2,026,634 ÷ 2 = 1,013,317 - No remainder! 2 is one of the factors!
1,013,317 ÷ 2 = 506,658.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,013,317 ÷ 3 = 337,772.3333 - This has a remainder. 3 is not a factor.
1,013,317 ÷ 5 = 202,663.4 - This has a remainder. 5 is not a factor.
1,013,317 ÷ 7 = 144,759.5714 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,013,317 ÷ 233 = 4,349 - No remainder! 233 is one of the factors!
4,349 ÷ 233 = 18.6652 - There is a remainder. We can't divide by 233 evenly anymore. Let's try the next prime number
4,349 ÷ 239 = 18.1967 - This has a remainder. 239 is not a factor.
4,349 ÷ 241 = 18.0456 - This has a remainder. 241 is not a factor.
4,349 ÷ 251 = 17.3267 - This has a remainder. 251 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,349 ÷ 4,349 = 1 - No remainder! 4,349 is one of the factors!

The orange divisor(s) above are the prime factors of the number 129,704,576. 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 233 x 4,349 = 129,704,576. It can also be written in exponential form as 27 x 2331 x 4,3491.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 129,704,576.

129,704,576
Factor Arrows
264,852,288
Factor Arrows
232,426,144
Factor Arrows
216,213,072
Factor Arrows
28,106,536
Factor Arrows
24,053,268
Factor Arrows
22,026,634
Factor Arrows
21,013,317
Factor Arrows
2334,349

More Prime Factorization Examples

129,704,574129,704,575129,704,577129,704,578
21 x 31 x 6311 x 34,259152 x 71 x 111 x 131 x 711 x 73131 x 171 x 7971 x 3,191121 x 4791 x 135,3911

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