Q: What is the prime factorization of the number 678,348?

 A:
  • The prime factors are: 2 x 2 x 3 x 3 x 3 x 11 x 571
    • or also written as { 2, 2, 3, 3, 3, 11, 571 }
  • Written in exponential form: 22 x 33 x 111 x 5711

Why is the prime factorization of 678,348 written as 22 x 33 x 111 x 5711?

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 678,348

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 678,348 by 2

678,348 ÷ 2 = 339,174 - No remainder! 2 is one of the factors!
339,174 ÷ 2 = 169,587 - No remainder! 2 is one of the factors!
169,587 ÷ 2 = 84,793.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
169,587 ÷ 3 = 56,529 - No remainder! 3 is one of the factors!
56,529 ÷ 3 = 18,843 - No remainder! 3 is one of the factors!
18,843 ÷ 3 = 6,281 - No remainder! 3 is one of the factors!
6,281 ÷ 3 = 2,093.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
6,281 ÷ 5 = 1,256.2 - This has a remainder. 5 is not a factor.
6,281 ÷ 7 = 897.2857 - This has a remainder. 7 is not a factor.
6,281 ÷ 11 = 571 - No remainder! 11 is one of the factors!
571 ÷ 11 = 51.9091 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
571 ÷ 13 = 43.9231 - This has a remainder. 13 is not a factor.
571 ÷ 17 = 33.5882 - This has a remainder. 17 is not a factor.
571 ÷ 19 = 30.0526 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
571 ÷ 571 = 1 - No remainder! 571 is one of the factors!

The orange divisor(s) above are the prime factors of the number 678,348. If we put all of it together we have the factors 2 x 2 x 3 x 3 x 3 x 11 x 571 = 678,348. It can also be written in exponential form as 22 x 33 x 111 x 5711.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 678,348.

678,348
Factor Arrows
2339,174
Factor Arrows
2169,587
Factor Arrows
356,529
Factor Arrows
318,843
Factor Arrows
36,281
Factor Arrows
11571

More Prime Factorization Examples

678,346678,347678,349678,350
21 x 339,1731531 x 12,799171 x 96,907121 x 52 x 13,5671

Try the factor calculator

Explore more about the number 678,348:


Ask a Question