Q: What is the prime factorization of the number 41,544,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 577
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 5, 5, 5, 577 }
  • Written in exponential form: 26 x 32 x 53 x 5771

Why is the prime factorization of 41,544,000 written as 26 x 32 x 53 x 5771?

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 41,544,000

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 41,544,000 by 2

41,544,000 ÷ 2 = 20,772,000 - No remainder! 2 is one of the factors!
20,772,000 ÷ 2 = 10,386,000 - No remainder! 2 is one of the factors!
10,386,000 ÷ 2 = 5,193,000 - No remainder! 2 is one of the factors!
5,193,000 ÷ 2 = 2,596,500 - No remainder! 2 is one of the factors!
2,596,500 ÷ 2 = 1,298,250 - No remainder! 2 is one of the factors!
1,298,250 ÷ 2 = 649,125 - No remainder! 2 is one of the factors!
649,125 ÷ 2 = 324,562.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
649,125 ÷ 3 = 216,375 - No remainder! 3 is one of the factors!
216,375 ÷ 3 = 72,125 - No remainder! 3 is one of the factors!
72,125 ÷ 3 = 24,041.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
72,125 ÷ 5 = 14,425 - No remainder! 5 is one of the factors!
14,425 ÷ 5 = 2,885 - No remainder! 5 is one of the factors!
2,885 ÷ 5 = 577 - No remainder! 5 is one of the factors!
577 ÷ 5 = 115.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
577 ÷ 7 = 82.4286 - This has a remainder. 7 is not a factor.
577 ÷ 11 = 52.4545 - This has a remainder. 11 is not a factor.
577 ÷ 13 = 44.3846 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
577 ÷ 577 = 1 - No remainder! 577 is one of the factors!

The orange divisor(s) above are the prime factors of the number 41,544,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 577 = 41,544,000. It can also be written in exponential form as 26 x 32 x 53 x 5771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 41,544,000.

41,544,000
Factor Arrows
220,772,000
Factor Arrows
210,386,000
Factor Arrows
25,193,000
Factor Arrows
22,596,500
Factor Arrows
21,298,250
Factor Arrows
2649,125
Factor Arrows
3216,375
Factor Arrows
372,125
Factor Arrows
514,425
Factor Arrows
52,885
Factor Arrows
5577

More Prime Factorization Examples

41,543,99841,543,99941,544,00141,544,002
21 x 1131 x 183,823171 x 311 x 191,44712111 x 4011 x 491121 x 3111 x 66,7911

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