Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
We cannot easily rephrase the question. Note that we may not need to know x in order to know how many factors it has.
Statement (1): INSUFFICIENT. Without knowing the value of n, we cannot determine the number of factors x has.
Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable n, whereas the question only involves the variable x.
Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of n.
Since n is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.
11 = 1 + 1? No.
22 = 2 + 2? Yes.
33 = 3 + 3? No.
44 = 4 + 4? No.
Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of n. Thus, we can determine that the value of n is 2.
Now, we do not know the value of p, nor of x, but we do now know that x = p2, with p as a prime number. Since a prime number has no factors other than 1 and itself, we can see that x has no factors other than 1, p, and p2. Thus, x has exactly 3 factors, and we can answer the question definitively.
The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.