Q: What is the prime factorization of the number 725,183?

 A:
  • The prime factors are: 31 x 149 x 157
    • or also written as { 31, 149, 157 }
  • Written in exponential form: 311 x 1491 x 1571

Why is the prime factorization of 725,183 written as 311 x 1491 x 1571?

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 725,183

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 725,183 by 2

725,183 ÷ 2 = 362,591.5 - This has a remainder. Let's try another prime number.
725,183 ÷ 3 = 241,727.6667 - This has a remainder. Let's try another prime number.
725,183 ÷ 5 = 145,036.6 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
725,183 ÷ 31 = 23,393 - No remainder! 31 is one of the factors!
23,393 ÷ 31 = 754.6129 - There is a remainder. We can't divide by 31 evenly anymore. Let's try the next prime number
23,393 ÷ 37 = 632.2432 - This has a remainder. 37 is not a factor.
23,393 ÷ 41 = 570.561 - This has a remainder. 41 is not a factor.
23,393 ÷ 43 = 544.0233 - This has a remainder. 43 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
23,393 ÷ 149 = 157 - No remainder! 149 is one of the factors!
157 ÷ 149 = 1.0537 - There is a remainder. We can't divide by 149 evenly anymore. Let's try the next prime number
157 ÷ 151 = 1.0397 - This has a remainder. 151 is not a factor.
157 ÷ 157 = 1 - No remainder! 157 is one of the factors!

The orange divisor(s) above are the prime factors of the number 725,183. If we put all of it together we have the factors 31 x 149 x 157 = 725,183. It can also be written in exponential form as 311 x 1491 x 1571.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 725,183.

725,183
Factor Arrows
3123,393
Factor Arrows
149157

More Prime Factorization Examples

725,181725,182725,184725,185
31 x 241,727121 x 731 x 4,967126 x 32 x 1,259151 x 145,0371

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