Hey everyone:
Rahul is right - the method he used is a somewhat obscure trick for "how many unique factors" questions. For a thorough breakdown you can check out a blog post I wrote here:
https://blog.veritasprep.com/2009/12/gma ... -week.html
That rule works essentially by using combinatorics logic - it's really a combinations problem (how many unique combinations of 3 ps and 6 qs can you create...plus there's also 1 which is a factor of everything). If you don't have the rule on this one - and it's obscure enough a rule that most GMAT takers won't have it memorized - you can work out the combinations pretty efficiently on your own. Keep in mind that, with the answer choices spread out the way they are, if you're close to the exact number you can be pretty certain you've got it.
Here's my breakdown (using 2^6 and 3^3 just because numbers seem more comfortable and easier to check):
1 (1 is a factor of all positive integers -
1 factor)
2^1 through 2^6 --->
6 factors
3^1 through 3^3 -->
3 factors
2*3
2*2*3
2*2*2*3
2*2*2*2*3
2*2*2*2*2*3
2*2*2*2*2*2*3 (kind of a pyramid structure...each potential number of 2s multiplied by one 3 - there are
6 of those)
Same thing as the above pyramid, but with two 3s: Another
6 unique factors
Same thing as the above with all three 3s: Another
6 unique factors
Add up all of those factors and you're at 28, which is choice D. If you can find just one more you'll have to pick E since D is too small, but we can't. Even if we start with 3s:
3*2 (already covered)
3*2*2 (already covered) etc.
We've already covered all of the combinations of 2s and 3s, so we know that the correct answer is 28.
If you're organized and look for opportunities to streamline the work (such as replicating that pyramid structure for one 3, two 3s, three 3s), finding 28 unique factors can be done in a minute or 90 seconds...remember, you don't care what the factors are, just that they exist, so you don't need to actually do the calculations.
