Q: What is the prime factorization of the number 41,035,420?

 A:
  • The prime factors are: 2 x 2 x 5 x 349 x 5,879
    • or also written as { 2, 2, 5, 349, 5,879 }
  • Written in exponential form: 22 x 51 x 3491 x 5,8791

Why is the prime factorization of 41,035,420 written as 22 x 51 x 3491 x 5,8791?

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 41,035,420

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 41,035,420 by 2

41,035,420 ÷ 2 = 20,517,710 - No remainder! 2 is one of the factors!
20,517,710 ÷ 2 = 10,258,855 - No remainder! 2 is one of the factors!
10,258,855 ÷ 2 = 5,129,427.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
10,258,855 ÷ 3 = 3,419,618.3333 - This has a remainder. 3 is not a factor.
10,258,855 ÷ 5 = 2,051,771 - No remainder! 5 is one of the factors!
2,051,771 ÷ 5 = 410,354.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
2,051,771 ÷ 7 = 293,110.1429 - This has a remainder. 7 is not a factor.
2,051,771 ÷ 11 = 186,524.6364 - This has a remainder. 11 is not a factor.
2,051,771 ÷ 13 = 157,828.5385 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,051,771 ÷ 349 = 5,879 - No remainder! 349 is one of the factors!
5,879 ÷ 349 = 16.8453 - There is a remainder. We can't divide by 349 evenly anymore. Let's try the next prime number
5,879 ÷ 353 = 16.6544 - This has a remainder. 353 is not a factor.
5,879 ÷ 359 = 16.376 - This has a remainder. 359 is not a factor.
5,879 ÷ 367 = 16.0191 - This has a remainder. 367 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
5,879 ÷ 5,879 = 1 - No remainder! 5,879 is one of the factors!

The orange divisor(s) above are the prime factors of the number 41,035,420. If we put all of it together we have the factors 2 x 2 x 5 x 349 x 5,879 = 41,035,420. It can also be written in exponential form as 22 x 51 x 3491 x 5,8791.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 41,035,420.

41,035,420
Factor Arrows
220,517,710
Factor Arrows
210,258,855
Factor Arrows
52,051,771
Factor Arrows
3495,879

More Prime Factorization Examples

41,035,41841,035,41941,035,42141,035,422
21 x 471 x 436,547132 x 4,559,491171 x 191 x 308,537121 x 31 x 6,839,2371

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