10^x + 10^y + 10^z = n, where x, y, and z are positive integers. Which

Every GMAT test includes a mix of standard questions (percent increase problems, solving equation problems, etc) and inventive questions that won't resemble too closely anything you've likely studied. The problem in this thread is of that second type -- I wouldn't really call it a "kind of GMAT question", because you might never see anything quite like it again. These inventive questions are hard to prepare for, because they truly test how well you can think about a brand new problem (and not whether you've memorized a formula or method), and you can't guess what concepts these problems will be testing. But if you're not sure how to proceed, often a good thing to do, if it's possible, is to try to generate some examples, to see if you can understand what's happening or notice a pattern.

Here we have this sum:

10^x + 10^y + 10^z

Each of these is a power of 10, so each number will look something like 100 or 1,000,000. If the powers are all different, so if we have something like:

10^5 + 10^3 + 10^1

then our sum will equal 100,000 + 1000 + 10 = 101,010, so it will just be a mix of 1's and 0's. If some of our exponents are identical, we might see a 2 or a 3 as one of our digits, and the rest will be zero, e.g.:

10^2 + 10^2 + 10^2 = 100 + 100 + 100 = 300

You might notice already that in this example, z = 2, and we have 2 zeros at the end of our sum, so z can be equal to the number of zeros, and III is possible. Again sticking with simple numbers, looking at item II, if we make y = 2 and z = 1, we want to get y - z = 1 zero at the end of our sum, and we can get that if x is 1 or 2:

10^2 + 10^2 + 10^1 = 210

so II is possible. Finally, looking at item I, if x = 1 and y = 1, we would want x + y = 2 zeros in our number. We'll always have one zero at the end of the number, so we want exactly one other zero as well, in the hundreds place in our number. We'll get that if z = 3:

10^1 + 10^1 + 10^3 = 1020

So I, II and III are all possible. It's a hard problem though, and not one I think is worth worrying about too much, though one thing you might notice is that we can solve it only by considering very simple examples -- we didn't need to make a single exponent greater than 3. In most similar GMAT questions, you don't need to think of anything too complicated to arrive at the right answer, but you do need to think carefully about all of the possibilities.