Explanation

We are asked whether . We could calculate the maximum possible value of m such that doesn't exceed 2,000 in advance of heading to the data statements, but they may give us a faster route, so we can head right to the data statements, separately first.

Statement (1) tells us that is, indeed, greater than 2,000. In one possible case, m could be quite large. In that case, would easily be greater than 2,000, and so would , and so the answer to the question would be "no." For a Case II, as we have already mentioned, there must be some value of m that is the maximum such that . If m is that value, then the next power of three will be over 2,000, in which case . So we can conclude that it's an allowed case that m is the maximum value such that , and in that case the answer to the question is "yes." With conflicting answers, we don't have sufficient information to answer the question definitively, and therefore Statement (1) is insufficient.

Statement (2) tells us information that is similar to what we were just imagining, because it describes the difference between one power of three and the next one up. We can list out powers of 3 to figure out which one is indicated here:







(I recommend memorizing at least the numbers, 3, 4, and 5 up to at least the 5th power.)



We can stop here, because . That means that , and therefore . Therefore we can answer the question definitively "yes." Statement (2) is sufficient and the correct answer is (B).

One bonus tidbit here: we have discussed before how, even when you are evaluating a statement separately, you can sometimes draw a hint from the other statements, such as a possible case. This question is a great example of a way to check your work on a data sufficiency question. Since the statements cannot contradict each other, you can check that after the fact. For example, if our analysis of Statement (2) had lead to the conclusion that m=5, that would have implied that we made a mistake, because then Statement (1) and Statement (2) would contradict each other. Meanwhile, we found , and , so in this case there is no contradiction.

Again, the correct answer is (B).