Q: What is the prime factorization of the number 10,325,312?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 161,333
    • or also written as { 2, 2, 2, 2, 2, 2, 161,333 }
  • Written in exponential form: 26 x 161,3331

Why is the prime factorization of 10,325,312 written as 26 x 161,3331?

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 10,325,312

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 10,325,312 by 2

10,325,312 ÷ 2 = 5,162,656 - No remainder! 2 is one of the factors!
5,162,656 ÷ 2 = 2,581,328 - No remainder! 2 is one of the factors!
2,581,328 ÷ 2 = 1,290,664 - No remainder! 2 is one of the factors!
1,290,664 ÷ 2 = 645,332 - No remainder! 2 is one of the factors!
645,332 ÷ 2 = 322,666 - No remainder! 2 is one of the factors!
322,666 ÷ 2 = 161,333 - No remainder! 2 is one of the factors!
161,333 ÷ 2 = 80,666.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
161,333 ÷ 3 = 53,777.6667 - This has a remainder. 3 is not a factor.
161,333 ÷ 5 = 32,266.6 - This has a remainder. 5 is not a factor.
161,333 ÷ 7 = 23,047.5714 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
161,333 ÷ 161,333 = 1 - No remainder! 161,333 is one of the factors!

The orange divisor(s) above are the prime factors of the number 10,325,312. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 161,333 = 10,325,312. It can also be written in exponential form as 26 x 161,3331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 10,325,312.

10,325,312
Factor Arrows
25,162,656
Factor Arrows
22,581,328
Factor Arrows
21,290,664
Factor Arrows
2645,332
Factor Arrows
2322,666
Factor Arrows
2161,333

More Prime Factorization Examples

10,325,31010,325,31110,325,31310,325,314
21 x 31 x 51 x 344,177110,325,311135 x 42,491121 x 1991 x 25,9431

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