What Defines a Number as Correct or Incorrect in This Mathematical Challenge?

[Impossiblebt]

Hi, my college professor gave me this brain teaser that I've been working at for a week but to no avail. Hopefully someone here can give me some assistance, because my professor won't.

Each natural number can be put into one of two categories: correct and incorrect. Each number will always be either incorrect or correct and will never change from one to the other.

Here are the examples he gave us: (note that the list of numbers is never-ending, he just gave us a small list)

Correct: 1-10, 12, 18, 20, 21, 24, 27, 30
Incorrect: 11, 13-17, 19, 22, 23, 25, 26, 28, 29

Over the past week, I've gotten to tell me at least this: there is no pattern, no sequence. Each number can be looked at individually and can be categorized within a few seconds, though longer numbers take a little longer to figure out (i.e. 4135). No calculator is necessary to do the math.
... To categorize each number, one needs to perform TWO math operations, that is +, -, x, or division. No complex math or big numbers involved. A sixth grader has the math knowledge to solve this. The result you get from doing these two operations will give you a number that has a certain number property. The property is what tells you which category it gets put into.

Good luck!

 

[Kaimyn]

I think I can see it!

A hint (if I'm correct, and none of the examples have failed):

Spoiler

You don't have to use any other, unrelated number on it (eg, you just don't (without a reason that relates this number to the original nubmer) divide by two). Just look at the number itself.

Another hint:

Spoiler

Any two digit multiple of 10 will be correct.

And the answer (if I'm correct):

Spoiler

A number is "correct" if the number is divisible by the sum of its digits. Else it is incorrect.