Q: What is the prime factorization of the number 337,663?

 A:
  • The prime factors are: 23 x 53 x 277
    • or also written as { 23, 53, 277 }
  • Written in exponential form: 231 x 531 x 2771

Why is the prime factorization of 337,663 written as 231 x 531 x 2771?

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 337,663

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 337,663 by 2

337,663 ÷ 2 = 168,831.5 - This has a remainder. Let's try another prime number.
337,663 ÷ 3 = 112,554.3333 - This has a remainder. Let's try another prime number.
337,663 ÷ 5 = 67,532.6 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
337,663 ÷ 23 = 14,681 - No remainder! 23 is one of the factors!
14,681 ÷ 23 = 638.3043 - There is a remainder. We can't divide by 23 evenly anymore. Let's try the next prime number
14,681 ÷ 29 = 506.2414 - This has a remainder. 29 is not a factor.
14,681 ÷ 31 = 473.5806 - This has a remainder. 31 is not a factor.
14,681 ÷ 37 = 396.7838 - This has a remainder. 37 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
14,681 ÷ 53 = 277 - No remainder! 53 is one of the factors!
277 ÷ 53 = 5.2264 - There is a remainder. We can't divide by 53 evenly anymore. Let's try the next prime number
277 ÷ 59 = 4.6949 - This has a remainder. 59 is not a factor.
277 ÷ 61 = 4.541 - This has a remainder. 61 is not a factor.
277 ÷ 67 = 4.1343 - This has a remainder. 67 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
277 ÷ 277 = 1 - No remainder! 277 is one of the factors!

The orange divisor(s) above are the prime factors of the number 337,663. If we put all of it together we have the factors 23 x 53 x 277 = 337,663. It can also be written in exponential form as 231 x 531 x 2771.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 337,663.

337,663
Factor Arrows
2314,681
Factor Arrows
53277

More Prime Factorization Examples

337,661337,662337,664337,665
337,661121 x 33 x 132 x 37128 x 1,319131 x 51 x 22,5111

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