Q: What is the prime factorization of the number 268,800?

 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 5 x 5 x 7
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 7 }
  • Written in exponential form: 29 x 31 x 52 x 71

Why is the prime factorization of 268,800 written as 29 x 31 x 52 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 268,800

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 268,800 by 2

268,800 ÷ 2 = 134,400 - No remainder! 2 is one of the factors!
134,400 ÷ 2 = 67,200 - No remainder! 2 is one of the factors!
67,200 ÷ 2 = 33,600 - No remainder! 2 is one of the factors!
33,600 ÷ 2 = 16,800 - No remainder! 2 is one of the factors!
16,800 ÷ 2 = 8,400 - No remainder! 2 is one of the factors!
8,400 ÷ 2 = 4,200 - No remainder! 2 is one of the factors!
4,200 ÷ 2 = 2,100 - No remainder! 2 is one of the factors!
2,100 ÷ 2 = 1,050 - No remainder! 2 is one of the factors!
1,050 ÷ 2 = 525 - No remainder! 2 is one of the factors!
525 ÷ 2 = 262.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
525 ÷ 3 = 175 - No remainder! 3 is one of the factors!
175 ÷ 3 = 58.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
175 ÷ 5 = 35 - No remainder! 5 is one of the factors!
35 ÷ 5 = 7 - No remainder! 5 is one of the factors!
7 ÷ 5 = 1.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
7 ÷ 7 = 1 - No remainder! 7 is one of the factors!

The orange divisor(s) above are the prime factors of the number 268,800. 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 5 x 5 x 7 = 268,800. It can also be written in exponential form as 29 x 31 x 52 x 71.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 268,800.

268,800
Factor Arrows
2134,400
Factor Arrows
267,200
Factor Arrows
233,600
Factor Arrows
216,800
Factor Arrows
28,400
Factor Arrows
24,200
Factor Arrows
22,100
Factor Arrows
21,050
Factor Arrows
2525
Factor Arrows
3175
Factor Arrows
535
Factor Arrows
57

More Prime Factorization Examples

268,798268,799268,801268,802
21 x 134,39913891 x 6911131 x 231 x 291 x 31121 x 134,4011

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