Manhattan GMAT Challenge Problem of the Week - 11 Dec 09

by , Dec 11, 2009

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 750+ level question, so do not worry if you cannot solve the problem in a 2 minute time frame. If you are up for the challenge, however, set your timer for 2 minutes and go!

Question

How many factors does x have, if x is a positive integer?

(1) x = [pmath]p^n[/pmath], where p is a prime number.

(2) [pmath]n^n[/pmath] = n + n, where n is a positive integer.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Solution

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.

[pmath]1^1[/pmath] = 1 + 1? No.

[pmath]2^2[/pmath] = 2 + 2? Yes.

[pmath]3^3[/pmath] = 3 + 3? No.

[pmath]4^4[/pmath] = 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 = [pmath]p^2[/pmath], 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 [pmath]p^2[/pmath]. 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.

To view the current Challenge Problem, simply visit the Challenge Problem page on Manhattan GMAT's website.