Q: What is the prime factorization of the number 578,169?

 A:
  • The prime factors are: 3 x 3 x 227 x 283
    • or also written as { 3, 3, 227, 283 }
  • Written in exponential form: 32 x 2271 x 2831

Why is the prime factorization of 578,169 written as 32 x 2271 x 2831?

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 578,169

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 578,169 by 2

578,169 ÷ 2 = 289,084.5 - This has a remainder. Let's try another prime number.
578,169 ÷ 3 = 192,723 - No remainder! 3 is one of the factors!
192,723 ÷ 3 = 64,241 - No remainder! 3 is one of the factors!
64,241 ÷ 3 = 21,413.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
64,241 ÷ 5 = 12,848.2 - This has a remainder. 5 is not a factor.
64,241 ÷ 7 = 9,177.2857 - This has a remainder. 7 is not a factor.
64,241 ÷ 11 = 5,840.0909 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
64,241 ÷ 227 = 283 - No remainder! 227 is one of the factors!
283 ÷ 227 = 1.2467 - There is a remainder. We can't divide by 227 evenly anymore. Let's try the next prime number
283 ÷ 229 = 1.2358 - This has a remainder. 229 is not a factor.
283 ÷ 233 = 1.2146 - This has a remainder. 233 is not a factor.
283 ÷ 239 = 1.1841 - This has a remainder. 239 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
283 ÷ 283 = 1 - No remainder! 283 is one of the factors!

The orange divisor(s) above are the prime factors of the number 578,169. If we put all of it together we have the factors 3 x 3 x 227 x 283 = 578,169. It can also be written in exponential form as 32 x 2271 x 2831.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 578,169.

578,169
Factor Arrows
3192,723
Factor Arrows
364,241
Factor Arrows
227283

More Prime Factorization Examples

578,167578,168578,170578,171
578,167123 x 72,271121 x 51 x 171 x 191 x 1791111 x 52,5611

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