Q: What is the prime factorization of the number 6,967,296?

 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 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 7 }
  • Written in exponential form: 212 x 35 x 71

Why is the prime factorization of 6,967,296 written as 212 x 35 x 71?

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 6,967,296

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 6,967,296 by 2

6,967,296 ÷ 2 = 3,483,648 - No remainder! 2 is one of the factors!
3,483,648 ÷ 2 = 1,741,824 - No remainder! 2 is one of the factors!
1,741,824 ÷ 2 = 870,912 - No remainder! 2 is one of the factors!
870,912 ÷ 2 = 435,456 - No remainder! 2 is one of the factors!
435,456 ÷ 2 = 217,728 - No remainder! 2 is one of the factors!
217,728 ÷ 2 = 108,864 - No remainder! 2 is one of the factors!
108,864 ÷ 2 = 54,432 - No remainder! 2 is one of the factors!
54,432 ÷ 2 = 27,216 - No remainder! 2 is one of the factors!
27,216 ÷ 2 = 13,608 - No remainder! 2 is one of the factors!
13,608 ÷ 2 = 6,804 - No remainder! 2 is one of the factors!
6,804 ÷ 2 = 3,402 - No remainder! 2 is one of the factors!
3,402 ÷ 2 = 1,701 - No remainder! 2 is one of the factors!
1,701 ÷ 2 = 850.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,701 ÷ 3 = 567 - No remainder! 3 is one of the factors!
567 ÷ 3 = 189 - No remainder! 3 is one of the factors!
189 ÷ 3 = 63 - No remainder! 3 is one of the factors!
63 ÷ 3 = 21 - No remainder! 3 is one of the factors!
21 ÷ 3 = 7 - No remainder! 3 is one of the factors!
7 ÷ 3 = 2.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
7 ÷ 5 = 1.4 - This has a remainder. 5 is not a factor.
7 ÷ 7 = 1 - No remainder! 7 is one of the factors!

The orange divisor(s) above are the prime factors of the number 6,967,296. 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 2 x 2 x 3 x 3 x 3 x 3 x 3 x 7 = 6,967,296. It can also be written in exponential form as 212 x 35 x 71.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 6,967,296.

6,967,296
Factor Arrows
23,483,648
Factor Arrows
21,741,824
Factor Arrows
2870,912
Factor Arrows
2435,456
Factor Arrows
2217,728
Factor Arrows
2108,864
Factor Arrows
254,432
Factor Arrows
227,216
Factor Arrows
213,608
Factor Arrows
26,804
Factor Arrows
23,402
Factor Arrows
21,701
Factor Arrows
3567
Factor Arrows
3189
Factor Arrows
363
Factor Arrows
321
Factor Arrows
37

More Prime Factorization Examples

6,967,2946,967,2956,967,2976,967,298
21 x 411 x 84,967151 x 1,393,4591171 x 409,841121 x 131 x 231 x 611 x 1911

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