Q: What is the prime factorization of the number 13,423,104?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 971
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 971 }
  • Written in exponential form: 29 x 33 x 9711

Why is the prime factorization of 13,423,104 written as 29 x 33 x 9711?

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 13,423,104

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 13,423,104 by 2

13,423,104 ÷ 2 = 6,711,552 - No remainder! 2 is one of the factors!
6,711,552 ÷ 2 = 3,355,776 - No remainder! 2 is one of the factors!
3,355,776 ÷ 2 = 1,677,888 - No remainder! 2 is one of the factors!
1,677,888 ÷ 2 = 838,944 - No remainder! 2 is one of the factors!
838,944 ÷ 2 = 419,472 - No remainder! 2 is one of the factors!
419,472 ÷ 2 = 209,736 - No remainder! 2 is one of the factors!
209,736 ÷ 2 = 104,868 - No remainder! 2 is one of the factors!
104,868 ÷ 2 = 52,434 - No remainder! 2 is one of the factors!
52,434 ÷ 2 = 26,217 - No remainder! 2 is one of the factors!
26,217 ÷ 2 = 13,108.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
26,217 ÷ 3 = 8,739 - No remainder! 3 is one of the factors!
8,739 ÷ 3 = 2,913 - No remainder! 3 is one of the factors!
2,913 ÷ 3 = 971 - No remainder! 3 is one of the factors!
971 ÷ 3 = 323.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
971 ÷ 5 = 194.2 - This has a remainder. 5 is not a factor.
971 ÷ 7 = 138.7143 - This has a remainder. 7 is not a factor.
971 ÷ 11 = 88.2727 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
971 ÷ 971 = 1 - No remainder! 971 is one of the factors!

The orange divisor(s) above are the prime factors of the number 13,423,104. 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 3 x 3 x 3 x 971 = 13,423,104. It can also be written in exponential form as 29 x 33 x 9711.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 13,423,104.

13,423,104
Factor Arrows
26,711,552
Factor Arrows
23,355,776
Factor Arrows
21,677,888
Factor Arrows
2838,944
Factor Arrows
2419,472
Factor Arrows
2209,736
Factor Arrows
2104,868
Factor Arrows
252,434
Factor Arrows
226,217
Factor Arrows
38,739
Factor Arrows
32,913
Factor Arrows
3971

More Prime Factorization Examples

13,423,10213,423,10313,423,10513,423,106
21 x 71 x 111 x 1011 x 863113,423,103151 x 1,2791 x 2,099121 x 471 x 142,7991

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