Q: What is the prime factorization of the number 11,205,120?

 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 5 x 1,459
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 1,459 }
  • Written in exponential form: 29 x 31 x 51 x 1,4591

Why is the prime factorization of 11,205,120 written as 29 x 31 x 51 x 1,4591?

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 11,205,120

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 11,205,120 by 2

11,205,120 ÷ 2 = 5,602,560 - No remainder! 2 is one of the factors!
5,602,560 ÷ 2 = 2,801,280 - No remainder! 2 is one of the factors!
2,801,280 ÷ 2 = 1,400,640 - No remainder! 2 is one of the factors!
1,400,640 ÷ 2 = 700,320 - No remainder! 2 is one of the factors!
700,320 ÷ 2 = 350,160 - No remainder! 2 is one of the factors!
350,160 ÷ 2 = 175,080 - No remainder! 2 is one of the factors!
175,080 ÷ 2 = 87,540 - No remainder! 2 is one of the factors!
87,540 ÷ 2 = 43,770 - No remainder! 2 is one of the factors!
43,770 ÷ 2 = 21,885 - No remainder! 2 is one of the factors!
21,885 ÷ 2 = 10,942.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
21,885 ÷ 3 = 7,295 - No remainder! 3 is one of the factors!
7,295 ÷ 3 = 2,431.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
7,295 ÷ 5 = 1,459 - No remainder! 5 is one of the factors!
1,459 ÷ 5 = 291.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,459 ÷ 7 = 208.4286 - This has a remainder. 7 is not a factor.
1,459 ÷ 11 = 132.6364 - This has a remainder. 11 is not a factor.
1,459 ÷ 13 = 112.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,459 ÷ 1,459 = 1 - No remainder! 1,459 is one of the factors!

The orange divisor(s) above are the prime factors of the number 11,205,120. 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 5 x 1,459 = 11,205,120. It can also be written in exponential form as 29 x 31 x 51 x 1,4591.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 11,205,120.

11,205,120
Factor Arrows
25,602,560
Factor Arrows
22,801,280
Factor Arrows
21,400,640
Factor Arrows
2700,320
Factor Arrows
2350,160
Factor Arrows
2175,080
Factor Arrows
287,540
Factor Arrows
243,770
Factor Arrows
221,885
Factor Arrows
37,295
Factor Arrows
51,459

More Prime Factorization Examples

11,205,11811,205,11911,205,12111,205,122
21 x 5,602,559111,205,119111,205,121121 x 5,602,5611

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