Q: What is the prime factorization of the number 972,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 5 x 5 x 5
    • or also written as { 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5 }
  • Written in exponential form: 25 x 35 x 53

Why is the prime factorization of 972,000 written as 25 x 35 x 53?

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 972,000

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 972,000 by 2

972,000 ÷ 2 = 486,000 - No remainder! 2 is one of the factors!
486,000 ÷ 2 = 243,000 - No remainder! 2 is one of the factors!
243,000 ÷ 2 = 121,500 - No remainder! 2 is one of the factors!
121,500 ÷ 2 = 60,750 - No remainder! 2 is one of the factors!
60,750 ÷ 2 = 30,375 - No remainder! 2 is one of the factors!
30,375 ÷ 2 = 15,187.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
30,375 ÷ 3 = 10,125 - No remainder! 3 is one of the factors!
10,125 ÷ 3 = 3,375 - No remainder! 3 is one of the factors!
3,375 ÷ 3 = 1,125 - No remainder! 3 is one of the factors!
1,125 ÷ 3 = 375 - No remainder! 3 is one of the factors!
375 ÷ 3 = 125 - No remainder! 3 is one of the factors!
125 ÷ 3 = 41.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
125 ÷ 5 = 25 - No remainder! 5 is one of the factors!
25 ÷ 5 = 5 - No remainder! 5 is one of the factors!
5 ÷ 5 = 1 - No remainder! 5 is one of the factors!

The orange divisor(s) above are the prime factors of the number 972,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 5 x 5 x 5 = 972,000. It can also be written in exponential form as 25 x 35 x 53.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 972,000.

972,000
Factor Arrows
2486,000
Factor Arrows
2243,000
Factor Arrows
2121,500
Factor Arrows
260,750
Factor Arrows
230,375
Factor Arrows
310,125
Factor Arrows
33,375
Factor Arrows
31,125
Factor Arrows
3375
Factor Arrows
3125
Factor Arrows
525
Factor Arrows
55

More Prime Factorization Examples

971,998971,999972,001972,002
21 x 1971 x 2,467171 x 1911 x 7271972,001121 x 191 x 25,5791

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