Q: What is the prime factorization of the number 398,545?

 A:
  • The prime factors are: 5 x 7 x 59 x 193
    • or also written as { 5, 7, 59, 193 }
  • Written in exponential form: 51 x 71 x 591 x 1931

Why is the prime factorization of 398,545 written as 51 x 71 x 591 x 1931?

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 398,545

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 398,545 by 2

398,545 ÷ 2 = 199,272.5 - This has a remainder. Let's try another prime number.
398,545 ÷ 3 = 132,848.3333 - This has a remainder. Let's try another prime number.
398,545 ÷ 5 = 79,709 - No remainder! 5 is one of the factors!
79,709 ÷ 5 = 15,941.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
79,709 ÷ 7 = 11,387 - No remainder! 7 is one of the factors!
11,387 ÷ 7 = 1,626.7143 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
11,387 ÷ 11 = 1,035.1818 - This has a remainder. 11 is not a factor.
11,387 ÷ 13 = 875.9231 - This has a remainder. 13 is not a factor.
11,387 ÷ 17 = 669.8235 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
11,387 ÷ 59 = 193 - No remainder! 59 is one of the factors!
193 ÷ 59 = 3.2712 - There is a remainder. We can't divide by 59 evenly anymore. Let's try the next prime number
193 ÷ 61 = 3.1639 - This has a remainder. 61 is not a factor.
193 ÷ 67 = 2.8806 - This has a remainder. 67 is not a factor.
193 ÷ 71 = 2.7183 - This has a remainder. 71 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
193 ÷ 193 = 1 - No remainder! 193 is one of the factors!

The orange divisor(s) above are the prime factors of the number 398,545. If we put all of it together we have the factors 5 x 7 x 59 x 193 = 398,545. It can also be written in exponential form as 51 x 71 x 591 x 1931.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 398,545.

398,545
Factor Arrows
579,709
Factor Arrows
711,387
Factor Arrows
59193

More Prime Factorization Examples

398,543398,544398,546398,547
398,543124 x 31 x 192 x 23121 x 1011 x 1,973133 x 291 x 5091
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