Q: What is the prime factorization of the number 59,340?

 A:
  • The prime factors are: 2 x 2 x 3 x 5 x 23 x 43
    • or also written as { 2, 2, 3, 5, 23, 43 }
  • Written in exponential form: 22 x 31 x 51 x 231 x 431

Why is the prime factorization of 59,340 written as 22 x 31 x 51 x 231 x 431?

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 59,340

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 59,340 by 2

59,340 ÷ 2 = 29,670 - No remainder! 2 is one of the factors!
29,670 ÷ 2 = 14,835 - No remainder! 2 is one of the factors!
14,835 ÷ 2 = 7,417.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
14,835 ÷ 3 = 4,945 - No remainder! 3 is one of the factors!
4,945 ÷ 3 = 1,648.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
4,945 ÷ 5 = 989 - No remainder! 5 is one of the factors!
989 ÷ 5 = 197.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
989 ÷ 7 = 141.2857 - This has a remainder. 7 is not a factor.
989 ÷ 11 = 89.9091 - This has a remainder. 11 is not a factor.
989 ÷ 13 = 76.0769 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
989 ÷ 23 = 43 - No remainder! 23 is one of the factors!
43 ÷ 23 = 1.8696 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
43 ÷ 29 = 1.4828 - This has a remainder. 29 is not a factor.
43 ÷ 31 = 1.3871 - This has a remainder. 31 is not a factor.
43 ÷ 37 = 1.1622 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
43 ÷ 43 = 1 - No remainder! 43 is one of the factors!

The orange divisor(s) above are the prime factors of the number 59,340. If we put all of it together we have the factors 2 x 2 x 3 x 5 x 23 x 43 = 59,340. It can also be written in exponential form as 22 x 31 x 51 x 231 x 431.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 59,340.

59,340
Factor Arrows
229,670
Factor Arrows
214,835
Factor Arrows
34,945
Factor Arrows
5989
Factor Arrows
2343

More Prime Factorization Examples

59,33859,33959,34159,342
21 x 29,669173 x 173159,341121 x 29,6711

Try the factor calculator

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