Q: What is the prime factorization of the number 1,005,312?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 11 x 17
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 11, 17 }
  • Written in exponential form: 28 x 31 x 71 x 111 x 171

Why is the prime factorization of 1,005,312 written as 28 x 31 x 71 x 111 x 171?

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 1,005,312

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 1,005,312 by 2

1,005,312 ÷ 2 = 502,656 - No remainder! 2 is one of the factors!
502,656 ÷ 2 = 251,328 - No remainder! 2 is one of the factors!
251,328 ÷ 2 = 125,664 - No remainder! 2 is one of the factors!
125,664 ÷ 2 = 62,832 - No remainder! 2 is one of the factors!
62,832 ÷ 2 = 31,416 - No remainder! 2 is one of the factors!
31,416 ÷ 2 = 15,708 - No remainder! 2 is one of the factors!
15,708 ÷ 2 = 7,854 - No remainder! 2 is one of the factors!
7,854 ÷ 2 = 3,927 - No remainder! 2 is one of the factors!
3,927 ÷ 2 = 1,963.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
3,927 ÷ 3 = 1,309 - No remainder! 3 is one of the factors!
1,309 ÷ 3 = 436.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,309 ÷ 5 = 261.8 - This has a remainder. 5 is not a factor.
1,309 ÷ 7 = 187 - No remainder! 7 is one of the factors!
187 ÷ 7 = 26.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
187 ÷ 11 = 17 - No remainder! 11 is one of the factors!
17 ÷ 11 = 1.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
17 ÷ 13 = 1.3077 - This has a remainder. 13 is not a factor.
17 ÷ 17 = 1 - No remainder! 17 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,005,312. 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 3 x 7 x 11 x 17 = 1,005,312. It can also be written in exponential form as 28 x 31 x 71 x 111 x 171.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,005,312.

1,005,312
Factor Arrows
2502,656
Factor Arrows
2251,328
Factor Arrows
2125,664
Factor Arrows
262,832
Factor Arrows
231,416
Factor Arrows
215,708
Factor Arrows
27,854
Factor Arrows
23,927
Factor Arrows
31,309
Factor Arrows
7187
Factor Arrows
1117

More Prime Factorization Examples

1,005,3101,005,3111,005,3131,005,314
21 x 51 x 2291 x 43914491 x 2,23911,005,313121 x 291 x 17,3331

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