Q: What is the prime factorization of the number 493,125?

 A:
  • The prime factors are: 3 x 5 x 5 x 5 x 5 x 263
    • or also written as { 3, 5, 5, 5, 5, 263 }
  • Written in exponential form: 31 x 54 x 2631

Why is the prime factorization of 493,125 written as 31 x 54 x 2631?

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 493,125

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 493,125 by 2

493,125 ÷ 2 = 246,562.5 - This has a remainder. Let's try another prime number.
493,125 ÷ 3 = 164,375 - No remainder! 3 is one of the factors!
164,375 ÷ 3 = 54,791.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
164,375 ÷ 5 = 32,875 - No remainder! 5 is one of the factors!
32,875 ÷ 5 = 6,575 - No remainder! 5 is one of the factors!
6,575 ÷ 5 = 1,315 - No remainder! 5 is one of the factors!
1,315 ÷ 5 = 263 - No remainder! 5 is one of the factors!
263 ÷ 5 = 52.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
263 ÷ 7 = 37.5714 - This has a remainder. 7 is not a factor.
263 ÷ 11 = 23.9091 - This has a remainder. 11 is not a factor.
263 ÷ 13 = 20.2308 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
263 ÷ 263 = 1 - No remainder! 263 is one of the factors!

The orange divisor(s) above are the prime factors of the number 493,125. If we put all of it together we have the factors 3 x 5 x 5 x 5 x 5 x 263 = 493,125. It can also be written in exponential form as 31 x 54 x 2631.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 493,125.

493,125
Factor Arrows
3164,375
Factor Arrows
532,875
Factor Arrows
56,575
Factor Arrows
51,315
Factor Arrows
5263

More Prime Factorization Examples

493,123493,124493,126493,127
493,123122 x 431 x 471 x 61121 x 192 x 6831493,1271

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