Q: What is the prime factorization of the number 3,245,184?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 313
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 313 }
  • Written in exponential form: 27 x 34 x 3131

Why is the prime factorization of 3,245,184 written as 27 x 34 x 3131?

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 3,245,184

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 3,245,184 by 2

3,245,184 ÷ 2 = 1,622,592 - No remainder! 2 is one of the factors!
1,622,592 ÷ 2 = 811,296 - No remainder! 2 is one of the factors!
811,296 ÷ 2 = 405,648 - No remainder! 2 is one of the factors!
405,648 ÷ 2 = 202,824 - No remainder! 2 is one of the factors!
202,824 ÷ 2 = 101,412 - No remainder! 2 is one of the factors!
101,412 ÷ 2 = 50,706 - No remainder! 2 is one of the factors!
50,706 ÷ 2 = 25,353 - No remainder! 2 is one of the factors!
25,353 ÷ 2 = 12,676.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
25,353 ÷ 3 = 8,451 - No remainder! 3 is one of the factors!
8,451 ÷ 3 = 2,817 - No remainder! 3 is one of the factors!
2,817 ÷ 3 = 939 - No remainder! 3 is one of the factors!
939 ÷ 3 = 313 - No remainder! 3 is one of the factors!
313 ÷ 3 = 104.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
313 ÷ 5 = 62.6 - This has a remainder. 5 is not a factor.
313 ÷ 7 = 44.7143 - This has a remainder. 7 is not a factor.
313 ÷ 11 = 28.4545 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
313 ÷ 313 = 1 - No remainder! 313 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,245,184. 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 3 x 3 x 3 x 3 x 313 = 3,245,184. It can also be written in exponential form as 27 x 34 x 3131.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,245,184.

3,245,184
Factor Arrows
21,622,592
Factor Arrows
2811,296
Factor Arrows
2405,648
Factor Arrows
2202,824
Factor Arrows
2101,412
Factor Arrows
250,706
Factor Arrows
225,353
Factor Arrows
38,451
Factor Arrows
32,817
Factor Arrows
3939
Factor Arrows
3313

More Prime Factorization Examples

3,245,1823,245,1833,245,1853,245,186
21 x 1,622,59114671 x 6,949151 x 231 x 28,219121 x 71 x 231,7991

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