Q: What is the prime factorization of the number 5,220,144?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 36,251
    • or also written as { 2, 2, 2, 2, 3, 3, 36,251 }
  • Written in exponential form: 24 x 32 x 36,2511

Why is the prime factorization of 5,220,144 written as 24 x 32 x 36,2511?

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 5,220,144

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 5,220,144 by 2

5,220,144 ÷ 2 = 2,610,072 - No remainder! 2 is one of the factors!
2,610,072 ÷ 2 = 1,305,036 - No remainder! 2 is one of the factors!
1,305,036 ÷ 2 = 652,518 - No remainder! 2 is one of the factors!
652,518 ÷ 2 = 326,259 - No remainder! 2 is one of the factors!
326,259 ÷ 2 = 163,129.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
326,259 ÷ 3 = 108,753 - No remainder! 3 is one of the factors!
108,753 ÷ 3 = 36,251 - No remainder! 3 is one of the factors!
36,251 ÷ 3 = 12,083.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
36,251 ÷ 5 = 7,250.2 - This has a remainder. 5 is not a factor.
36,251 ÷ 7 = 5,178.7143 - This has a remainder. 7 is not a factor.
36,251 ÷ 11 = 3,295.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
36,251 ÷ 36,251 = 1 - No remainder! 36,251 is one of the factors!

The orange divisor(s) above are the prime factors of the number 5,220,144. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 36,251 = 5,220,144. It can also be written in exponential form as 24 x 32 x 36,2511.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 5,220,144.

5,220,144
Factor Arrows
22,610,072
Factor Arrows
21,305,036
Factor Arrows
2652,518
Factor Arrows
2326,259
Factor Arrows
3108,753
Factor Arrows
336,251

More Prime Factorization Examples

5,220,1425,220,1435,220,1455,220,146
21 x 4791 x 5,4491591 x 1031 x 859151 x 71 x 291 x 371 x 139121 x 1,3731 x 1,9011

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