Q: What is the prime factorization of the number 41,821,551?

 A:
  • The prime factors are: 3 x 3 x 191 x 24,329
    • or also written as { 3, 3, 191, 24,329 }
  • Written in exponential form: 32 x 1911 x 24,3291

Why is the prime factorization of 41,821,551 written as 32 x 1911 x 24,3291?

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 41,821,551

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 41,821,551 by 2

41,821,551 ÷ 2 = 20,910,775.5 - This has a remainder. Let's try another prime number.
41,821,551 ÷ 3 = 13,940,517 - No remainder! 3 is one of the factors!
13,940,517 ÷ 3 = 4,646,839 - No remainder! 3 is one of the factors!
4,646,839 ÷ 3 = 1,548,946.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
4,646,839 ÷ 5 = 929,367.8 - This has a remainder. 5 is not a factor.
4,646,839 ÷ 7 = 663,834.1429 - This has a remainder. 7 is not a factor.
4,646,839 ÷ 11 = 422,439.9091 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
4,646,839 ÷ 191 = 24,329 - No remainder! 191 is one of the factors!
24,329 ÷ 191 = 127.377 - There is a remainder. We can't divide by 191 evenly anymore. Let's try the next prime number
24,329 ÷ 193 = 126.057 - This has a remainder. 193 is not a factor.
24,329 ÷ 197 = 123.4975 - This has a remainder. 197 is not a factor.
24,329 ÷ 199 = 122.2563 - This has a remainder. 199 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,329 ÷ 24,329 = 1 - No remainder! 24,329 is one of the factors!

The orange divisor(s) above are the prime factors of the number 41,821,551. If we put all of it together we have the factors 3 x 3 x 191 x 24,329 = 41,821,551. It can also be written in exponential form as 32 x 1911 x 24,3291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 41,821,551.

41,821,551
Factor Arrows
313,940,517
Factor Arrows
34,646,839
Factor Arrows
19124,329

More Prime Factorization Examples

41,821,54941,821,55041,821,55241,821,553
72 x 111 x 77,591121 x 52 x 971 x 8,623124 x 2,613,847141,821,5531

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