Q: What is the prime factorization of the number 675,840?

 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 5 x 11
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 11 }
  • Written in exponential form: 212 x 31 x 51 x 111

Why is the prime factorization of 675,840 written as 212 x 31 x 51 x 111?

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 675,840

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 675,840 by 2

675,840 ÷ 2 = 337,920 - No remainder! 2 is one of the factors!
337,920 ÷ 2 = 168,960 - No remainder! 2 is one of the factors!
168,960 ÷ 2 = 84,480 - No remainder! 2 is one of the factors!
84,480 ÷ 2 = 42,240 - No remainder! 2 is one of the factors!
42,240 ÷ 2 = 21,120 - No remainder! 2 is one of the factors!
21,120 ÷ 2 = 10,560 - No remainder! 2 is one of the factors!
10,560 ÷ 2 = 5,280 - No remainder! 2 is one of the factors!
5,280 ÷ 2 = 2,640 - No remainder! 2 is one of the factors!
2,640 ÷ 2 = 1,320 - No remainder! 2 is one of the factors!
1,320 ÷ 2 = 660 - No remainder! 2 is one of the factors!
660 ÷ 2 = 330 - No remainder! 2 is one of the factors!
330 ÷ 2 = 165 - No remainder! 2 is one of the factors!
165 ÷ 2 = 82.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
165 ÷ 3 = 55 - No remainder! 3 is one of the factors!
55 ÷ 3 = 18.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
55 ÷ 5 = 11 - No remainder! 5 is one of the factors!
11 ÷ 5 = 2.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
11 ÷ 7 = 1.5714 - This has a remainder. 7 is not a factor.
11 ÷ 11 = 1 - No remainder! 11 is one of the factors!

The orange divisor(s) above are the prime factors of the number 675,840. 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 5 x 11 = 675,840. It can also be written in exponential form as 212 x 31 x 51 x 111.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 675,840.

675,840
Factor Arrows
2337,920
Factor Arrows
2168,960
Factor Arrows
284,480
Factor Arrows
242,240
Factor Arrows
221,120
Factor Arrows
210,560
Factor Arrows
25,280
Factor Arrows
22,640
Factor Arrows
21,320
Factor Arrows
2660
Factor Arrows
2330
Factor Arrows
2165
Factor Arrows
355
Factor Arrows
511

More Prime Factorization Examples

675,838675,839675,841675,842
21 x 337,9191675,8391675,841121 x 371 x 9,1331

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