Q: What is the prime factorization of the number 332,404,224?

 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 379 x 571
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 379, 571 }
  • Written in exponential form: 29 x 31 x 3791 x 5711

Why is the prime factorization of 332,404,224 written as 29 x 31 x 3791 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 332,404,224

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 332,404,224 by 2

332,404,224 ÷ 2 = 166,202,112 - No remainder! 2 is one of the factors!
166,202,112 ÷ 2 = 83,101,056 - No remainder! 2 is one of the factors!
83,101,056 ÷ 2 = 41,550,528 - No remainder! 2 is one of the factors!
41,550,528 ÷ 2 = 20,775,264 - No remainder! 2 is one of the factors!
20,775,264 ÷ 2 = 10,387,632 - No remainder! 2 is one of the factors!
10,387,632 ÷ 2 = 5,193,816 - No remainder! 2 is one of the factors!
5,193,816 ÷ 2 = 2,596,908 - No remainder! 2 is one of the factors!
2,596,908 ÷ 2 = 1,298,454 - No remainder! 2 is one of the factors!
1,298,454 ÷ 2 = 649,227 - No remainder! 2 is one of the factors!
649,227 ÷ 2 = 324,613.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
649,227 ÷ 3 = 216,409 - No remainder! 3 is one of the factors!
216,409 ÷ 3 = 72,136.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
216,409 ÷ 5 = 43,281.8 - This has a remainder. 5 is not a factor.
216,409 ÷ 7 = 30,915.5714 - This has a remainder. 7 is not a factor.
216,409 ÷ 11 = 19,673.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
216,409 ÷ 379 = 571 - No remainder! 379 is one of the factors!
571 ÷ 379 = 1.5066 - There is a remainder. We can't divide by 379 evenly anymore. Let's try the next prime number
571 ÷ 383 = 1.4909 - This has a remainder. 383 is not a factor.
571 ÷ 389 = 1.4679 - This has a remainder. 389 is not a factor.
571 ÷ 397 = 1.4383 - This has a remainder. 397 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 332,404,224. 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 379 x 571 = 332,404,224. It can also be written in exponential form as 29 x 31 x 3791 x 5711.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 332,404,224.

332,404,224
Factor Arrows
2166,202,112
Factor Arrows
283,101,056
Factor Arrows
241,550,528
Factor Arrows
220,775,264
Factor Arrows
210,387,632
Factor Arrows
25,193,816
Factor Arrows
22,596,908
Factor Arrows
21,298,454
Factor Arrows
2649,227
Factor Arrows
3216,409
Factor Arrows
379571

More Prime Factorization Examples

332,404,222332,404,223332,404,225332,404,226
21 x 166,202,11112,0111 x 165,293152 x 13,296,169121 x 71 x 111 x 371 x 58,3371

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