Q: What is the prime factorization of the number 41,261,312?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 17 x 19 x 499
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 17, 19, 499 }
  • Written in exponential form: 28 x 171 x 191 x 4991

Why is the prime factorization of 41,261,312 written as 28 x 171 x 191 x 4991?

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 41,261,312

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 41,261,312 by 2

41,261,312 ÷ 2 = 20,630,656 - No remainder! 2 is one of the factors!
20,630,656 ÷ 2 = 10,315,328 - No remainder! 2 is one of the factors!
10,315,328 ÷ 2 = 5,157,664 - No remainder! 2 is one of the factors!
5,157,664 ÷ 2 = 2,578,832 - No remainder! 2 is one of the factors!
2,578,832 ÷ 2 = 1,289,416 - No remainder! 2 is one of the factors!
1,289,416 ÷ 2 = 644,708 - No remainder! 2 is one of the factors!
644,708 ÷ 2 = 322,354 - No remainder! 2 is one of the factors!
322,354 ÷ 2 = 161,177 - No remainder! 2 is one of the factors!
161,177 ÷ 2 = 80,588.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
161,177 ÷ 3 = 53,725.6667 - This has a remainder. 3 is not a factor.
161,177 ÷ 5 = 32,235.4 - This has a remainder. 5 is not a factor.
161,177 ÷ 7 = 23,025.2857 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
161,177 ÷ 17 = 9,481 - No remainder! 17 is one of the factors!
9,481 ÷ 17 = 557.7059 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
9,481 ÷ 19 = 499 - No remainder! 19 is one of the factors!
499 ÷ 19 = 26.2632 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
499 ÷ 23 = 21.6957 - This has a remainder. 23 is not a factor.
499 ÷ 29 = 17.2069 - This has a remainder. 29 is not a factor.
499 ÷ 31 = 16.0968 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
499 ÷ 499 = 1 - No remainder! 499 is one of the factors!

The orange divisor(s) above are the prime factors of the number 41,261,312. 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 17 x 19 x 499 = 41,261,312. It can also be written in exponential form as 28 x 171 x 191 x 4991.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 41,261,312.

41,261,312
Factor Arrows
220,630,656
Factor Arrows
210,315,328
Factor Arrows
25,157,664
Factor Arrows
22,578,832
Factor Arrows
21,289,416
Factor Arrows
2644,708
Factor Arrows
2322,354
Factor Arrows
2161,177
Factor Arrows
179,481
Factor Arrows
19499

More Prime Factorization Examples

41,261,31041,261,31141,261,31341,261,314
21 x 32 x 51 x 231 x 311 x 643171 x 131 x 453,421131 x 13,753,771121 x 2,4471 x 8,4311

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