Q: What is the prime factorization of the number 242,414,144?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 7 x 191 x 2,833
    • or also written as { 2, 2, 2, 2, 2, 2, 7, 191, 2,833 }
  • Written in exponential form: 26 x 71 x 1911 x 2,8331

Why is the prime factorization of 242,414,144 written as 26 x 71 x 1911 x 2,8331?

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 242,414,144

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 242,414,144 by 2

242,414,144 ÷ 2 = 121,207,072 - No remainder! 2 is one of the factors!
121,207,072 ÷ 2 = 60,603,536 - No remainder! 2 is one of the factors!
60,603,536 ÷ 2 = 30,301,768 - No remainder! 2 is one of the factors!
30,301,768 ÷ 2 = 15,150,884 - No remainder! 2 is one of the factors!
15,150,884 ÷ 2 = 7,575,442 - No remainder! 2 is one of the factors!
7,575,442 ÷ 2 = 3,787,721 - No remainder! 2 is one of the factors!
3,787,721 ÷ 2 = 1,893,860.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
3,787,721 ÷ 3 = 1,262,573.6667 - This has a remainder. 3 is not a factor.
3,787,721 ÷ 5 = 757,544.2 - This has a remainder. 5 is not a factor.
3,787,721 ÷ 7 = 541,103 - No remainder! 7 is one of the factors!
541,103 ÷ 7 = 77,300.4286 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
541,103 ÷ 11 = 49,191.1818 - This has a remainder. 11 is not a factor.
541,103 ÷ 13 = 41,623.3077 - This has a remainder. 13 is not a factor.
541,103 ÷ 17 = 31,829.5882 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
541,103 ÷ 191 = 2,833 - No remainder! 191 is one of the factors!
2,833 ÷ 191 = 14.8325 - There is a remainder. We can't divide by 191 evenly anymore. Let's try the next prime number
2,833 ÷ 193 = 14.6788 - This has a remainder. 193 is not a factor.
2,833 ÷ 197 = 14.3807 - This has a remainder. 197 is not a factor.
2,833 ÷ 199 = 14.2362 - This has a remainder. 199 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
2,833 ÷ 2,833 = 1 - No remainder! 2,833 is one of the factors!

The orange divisor(s) above are the prime factors of the number 242,414,144. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 7 x 191 x 2,833 = 242,414,144. It can also be written in exponential form as 26 x 71 x 1911 x 2,8331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 242,414,144.

242,414,144
Factor Arrows
2121,207,072
Factor Arrows
260,603,536
Factor Arrows
230,301,768
Factor Arrows
215,150,884
Factor Arrows
27,575,442
Factor Arrows
23,787,721
Factor Arrows
7541,103
Factor Arrows
1912,833

More Prime Factorization Examples

242,414,142242,414,143242,414,145242,414,146
21 x 31 x 40,402,35713671 x 660,529132 x 51 x 5,386,981121 x 131 x 1,4831 x 6,2871

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