Woof - this is a case for modular arithmetic! The GMAT doesn't expect us to know what that is, however, so let's see if we can solve this with GMAT math.
S1::
Look for a pattern involving the tens digit of 11�.
11¹ = 11, 11² = 121, 11³ = 1331, ...
The tens digit appears to increase by 1 per power (and if you do the multiplication by hand, it'll be easy to see why). From this, we can conjecture that 11�, 11¹�, 11²�, etc. will all have a tens digit of 4. So n = 10k + 4, where k is some nonnegative integer. Insufficient.
S2::
Look for a pattern involving the hundreds digit of 5�. Here are the first few powers:
5, 25, 125, 625, 3125, 15625, ...
Now notice what's going to happen when we multiply by 5 again.
15625 * 5 = (15,000 + 625) * 5 = 15,000*5 + 625*5 = 75,000 + 3,125
So the hundreds digit again is one!
At this point, we can see the the next power will work the same way: only the last three digits (125) times 5 will influence the hundreds digit. Our powers will thus go ...125, ...625, ...125, ...625, forever onward, it seems.
So we can conjecture that 5�, for all even n greater than 2, will have a hundreds digit of 6. Insufficient.
S1 and S2::
All the options from S1 are even, so S2 doesn't help, and we can't answer the question.
Great problem!
