Q: What is the prime factorization of the number 324,784,640?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 293 x 433
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 293, 433 }
  • Written in exponential form: 29 x 51 x 2931 x 4331

Why is the prime factorization of 324,784,640 written as 29 x 51 x 2931 x 4331?

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 324,784,640

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 324,784,640 by 2

324,784,640 ÷ 2 = 162,392,320 - No remainder! 2 is one of the factors!
162,392,320 ÷ 2 = 81,196,160 - No remainder! 2 is one of the factors!
81,196,160 ÷ 2 = 40,598,080 - No remainder! 2 is one of the factors!
40,598,080 ÷ 2 = 20,299,040 - No remainder! 2 is one of the factors!
20,299,040 ÷ 2 = 10,149,520 - No remainder! 2 is one of the factors!
10,149,520 ÷ 2 = 5,074,760 - No remainder! 2 is one of the factors!
5,074,760 ÷ 2 = 2,537,380 - No remainder! 2 is one of the factors!
2,537,380 ÷ 2 = 1,268,690 - No remainder! 2 is one of the factors!
1,268,690 ÷ 2 = 634,345 - No remainder! 2 is one of the factors!
634,345 ÷ 2 = 317,172.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
634,345 ÷ 3 = 211,448.3333 - This has a remainder. 3 is not a factor.
634,345 ÷ 5 = 126,869 - No remainder! 5 is one of the factors!
126,869 ÷ 5 = 25,373.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
126,869 ÷ 7 = 18,124.1429 - This has a remainder. 7 is not a factor.
126,869 ÷ 11 = 11,533.5455 - This has a remainder. 11 is not a factor.
126,869 ÷ 13 = 9,759.1538 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
126,869 ÷ 293 = 433 - No remainder! 293 is one of the factors!
433 ÷ 293 = 1.4778 - There is a remainder. We can't divide by 293 evenly anymore. Let's try the next prime number
433 ÷ 307 = 1.4104 - This has a remainder. 307 is not a factor.
433 ÷ 311 = 1.3923 - This has a remainder. 311 is not a factor.
433 ÷ 313 = 1.3834 - This has a remainder. 313 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
433 ÷ 433 = 1 - No remainder! 433 is one of the factors!

The orange divisor(s) above are the prime factors of the number 324,784,640. 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 5 x 293 x 433 = 324,784,640. It can also be written in exponential form as 29 x 51 x 2931 x 4331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 324,784,640.

324,784,640
Factor Arrows
2162,392,320
Factor Arrows
281,196,160
Factor Arrows
240,598,080
Factor Arrows
220,299,040
Factor Arrows
210,149,520
Factor Arrows
25,074,760
Factor Arrows
22,537,380
Factor Arrows
21,268,690
Factor Arrows
2634,345
Factor Arrows
5126,869
Factor Arrows
293433

More Prime Factorization Examples

324,784,638324,784,639324,784,641324,784,642
21 x 32 x 18,043,5911324,784,639131 x 1971 x 549,551121 x 73 x 131 x 791 x 4611

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