Q: What is the prime factorization of the number 237,568?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 29
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 29 }
  • Written in exponential form: 213 x 291

Why is the prime factorization of 237,568 written as 213 x 291?

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 237,568

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 237,568 by 2

237,568 ÷ 2 = 118,784 - No remainder! 2 is one of the factors!
118,784 ÷ 2 = 59,392 - No remainder! 2 is one of the factors!
59,392 ÷ 2 = 29,696 - No remainder! 2 is one of the factors!
29,696 ÷ 2 = 14,848 - No remainder! 2 is one of the factors!
14,848 ÷ 2 = 7,424 - No remainder! 2 is one of the factors!
7,424 ÷ 2 = 3,712 - No remainder! 2 is one of the factors!
3,712 ÷ 2 = 1,856 - No remainder! 2 is one of the factors!
1,856 ÷ 2 = 928 - No remainder! 2 is one of the factors!
928 ÷ 2 = 464 - No remainder! 2 is one of the factors!
464 ÷ 2 = 232 - No remainder! 2 is one of the factors!
232 ÷ 2 = 116 - No remainder! 2 is one of the factors!
116 ÷ 2 = 58 - No remainder! 2 is one of the factors!
58 ÷ 2 = 29 - No remainder! 2 is one of the factors!
29 ÷ 2 = 14.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
29 ÷ 3 = 9.6667 - This has a remainder. 3 is not a factor.
29 ÷ 5 = 5.8 - This has a remainder. 5 is not a factor.
29 ÷ 7 = 4.1429 - This has a remainder. 7 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
29 ÷ 29 = 1 - No remainder! 29 is one of the factors!

The orange divisor(s) above are the prime factors of the number 237,568. 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 2 x 2 x 2 x 2 x 2 x 29 = 237,568. It can also be written in exponential form as 213 x 291.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 237,568.

237,568
Factor Arrows
2118,784
Factor Arrows
259,392
Factor Arrows
229,696
Factor Arrows
214,848
Factor Arrows
27,424
Factor Arrows
23,712
Factor Arrows
21,856
Factor Arrows
2928
Factor Arrows
2464
Factor Arrows
2232
Factor Arrows
2116
Factor Arrows
258
Factor Arrows
229

More Prime Factorization Examples

237,566237,567237,569237,570
21 x 71 x 711 x 239131 x 111 x 231 x 31313531 x 673121 x 31 x 51 x 7,9191

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