Q: What is the prime factorization of the number 721,312?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 22,541
    • or also written as { 2, 2, 2, 2, 2, 22,541 }
  • Written in exponential form: 25 x 22,5411

Why is the prime factorization of 721,312 written as 25 x 22,5411?

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 721,312

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 721,312 by 2

721,312 ÷ 2 = 360,656 - No remainder! 2 is one of the factors!
360,656 ÷ 2 = 180,328 - No remainder! 2 is one of the factors!
180,328 ÷ 2 = 90,164 - No remainder! 2 is one of the factors!
90,164 ÷ 2 = 45,082 - No remainder! 2 is one of the factors!
45,082 ÷ 2 = 22,541 - No remainder! 2 is one of the factors!
22,541 ÷ 2 = 11,270.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
22,541 ÷ 3 = 7,513.6667 - This has a remainder. 3 is not a factor.
22,541 ÷ 5 = 4,508.2 - This has a remainder. 5 is not a factor.
22,541 ÷ 7 = 3,220.1429 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
22,541 ÷ 22,541 = 1 - No remainder! 22,541 is one of the factors!

The orange divisor(s) above are the prime factors of the number 721,312. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 22,541 = 721,312. It can also be written in exponential form as 25 x 22,5411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 721,312.

721,312
Factor Arrows
2360,656
Factor Arrows
2180,328
Factor Arrows
290,164
Factor Arrows
245,082
Factor Arrows
222,541

More Prime Factorization Examples

721,310721,311721,313721,314
21 x 51 x 171 x 4,243131 x 240,4371411 x 731 x 241121 x 32 x 111 x 3,6431

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