Q: What is the prime factorization of the number 181,856,124?

 A:
  • The prime factors are: 2 x 2 x 3 x 3 x 3 x 23 x 179 x 409
    • or also written as { 2, 2, 3, 3, 3, 23, 179, 409 }
  • Written in exponential form: 22 x 33 x 231 x 1791 x 4091

Why is the prime factorization of 181,856,124 written as 22 x 33 x 231 x 1791 x 4091?

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 181,856,124

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 181,856,124 by 2

181,856,124 ÷ 2 = 90,928,062 - No remainder! 2 is one of the factors!
90,928,062 ÷ 2 = 45,464,031 - No remainder! 2 is one of the factors!
45,464,031 ÷ 2 = 22,732,015.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
45,464,031 ÷ 3 = 15,154,677 - No remainder! 3 is one of the factors!
15,154,677 ÷ 3 = 5,051,559 - No remainder! 3 is one of the factors!
5,051,559 ÷ 3 = 1,683,853 - No remainder! 3 is one of the factors!
1,683,853 ÷ 3 = 561,284.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,683,853 ÷ 5 = 336,770.6 - This has a remainder. 5 is not a factor.
1,683,853 ÷ 7 = 240,550.4286 - This has a remainder. 7 is not a factor.
1,683,853 ÷ 11 = 153,077.5455 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,683,853 ÷ 23 = 73,211 - No remainder! 23 is one of the factors!
73,211 ÷ 23 = 3,183.087 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
73,211 ÷ 29 = 2,524.5172 - This has a remainder. 29 is not a factor.
73,211 ÷ 31 = 2,361.6452 - This has a remainder. 31 is not a factor.
73,211 ÷ 37 = 1,978.6757 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
73,211 ÷ 179 = 409 - No remainder! 179 is one of the factors!
409 ÷ 179 = 2.2849 - There is a remainder. We can't divide by 179 evenly anymore. Let's try the next prime number
409 ÷ 181 = 2.2597 - This has a remainder. 181 is not a factor.
409 ÷ 191 = 2.1414 - This has a remainder. 191 is not a factor.
409 ÷ 193 = 2.1192 - This has a remainder. 193 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
409 ÷ 409 = 1 - No remainder! 409 is one of the factors!

The orange divisor(s) above are the prime factors of the number 181,856,124. If we put all of it together we have the factors 2 x 2 x 3 x 3 x 3 x 23 x 179 x 409 = 181,856,124. It can also be written in exponential form as 22 x 33 x 231 x 1791 x 4091.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 181,856,124.

181,856,124
Factor Arrows
290,928,062
Factor Arrows
245,464,031
Factor Arrows
315,154,677
Factor Arrows
35,051,559
Factor Arrows
31,683,853
Factor Arrows
2373,211
Factor Arrows
179409

More Prime Factorization Examples

181,856,122181,856,123181,856,125181,856,126
21 x 71 x 5871 x 22,1291171 x 10,697,419153 x 111 x 191 x 6,961121 x 1271 x 715,9691

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