Q: What is the prime factorization of the number 4,036,608?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 73
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 73 }
  • Written in exponential form: 211 x 33 x 731

Why is the prime factorization of 4,036,608 written as 211 x 33 x 731?

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 4,036,608

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 4,036,608 by 2

4,036,608 ÷ 2 = 2,018,304 - No remainder! 2 is one of the factors!
2,018,304 ÷ 2 = 1,009,152 - No remainder! 2 is one of the factors!
1,009,152 ÷ 2 = 504,576 - No remainder! 2 is one of the factors!
504,576 ÷ 2 = 252,288 - No remainder! 2 is one of the factors!
252,288 ÷ 2 = 126,144 - No remainder! 2 is one of the factors!
126,144 ÷ 2 = 63,072 - No remainder! 2 is one of the factors!
63,072 ÷ 2 = 31,536 - No remainder! 2 is one of the factors!
31,536 ÷ 2 = 15,768 - No remainder! 2 is one of the factors!
15,768 ÷ 2 = 7,884 - No remainder! 2 is one of the factors!
7,884 ÷ 2 = 3,942 - No remainder! 2 is one of the factors!
3,942 ÷ 2 = 1,971 - No remainder! 2 is one of the factors!
1,971 ÷ 2 = 985.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,971 ÷ 3 = 657 - No remainder! 3 is one of the factors!
657 ÷ 3 = 219 - No remainder! 3 is one of the factors!
219 ÷ 3 = 73 - No remainder! 3 is one of the factors!
73 ÷ 3 = 24.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
73 ÷ 5 = 14.6 - This has a remainder. 5 is not a factor.
73 ÷ 7 = 10.4286 - This has a remainder. 7 is not a factor.
73 ÷ 11 = 6.6364 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
73 ÷ 73 = 1 - No remainder! 73 is one of the factors!

The orange divisor(s) above are the prime factors of the number 4,036,608. 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 2 x 2 x 3 x 3 x 3 x 73 = 4,036,608. It can also be written in exponential form as 211 x 33 x 731.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 4,036,608.

4,036,608
Factor Arrows
22,018,304
Factor Arrows
21,009,152
Factor Arrows
2504,576
Factor Arrows
2252,288
Factor Arrows
2126,144
Factor Arrows
263,072
Factor Arrows
231,536
Factor Arrows
215,768
Factor Arrows
27,884
Factor Arrows
23,942
Factor Arrows
21,971
Factor Arrows
3657
Factor Arrows
3219
Factor Arrows
373

More Prime Factorization Examples

4,036,6064,036,6074,036,6094,036,610
21 x 71 x 3731 x 7731191 x 212,45311731 x 23,333121 x 51 x 403,6611

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