Here is the answer to yesterday’s Challenge Question, brought to you by Manhattan GMAT!
Question
This week’s challenge problem is a Data Sufficiency question.
If P, Q, R, and S are positive integers, and 
, is R divisible by 5 ?
(1) P is divisible by 140
(2) 
, where x 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.
Answer
Let’s begin by analyzing the information given to us in the question:
If P, Q, R, and S are positive integers, and , is R divisible by 5 ?It is often helpful on the GMAT to rephrase equations so that there are no denominators. We can do this my cross-multiplying as follows:
Now let’s analyze Statement (1) alone: P is divisible by 140.
Most GMAT divisibility problems can be solved by breaking numbers down to their prime factors (this is called a “prime factorization”).
The prime factorization of 140 is: 
.
Thus, if P is divisible by 140, it is also divisible by all the prime factors of 140. We know that P is divisible by 2 twice, by 5, and by 7. However, this gives us no information about R so Statement (1) is not sufficient to answer the question.
Next, let’s analyze Statement (2) alone: , where x is a positive integer.
From this, we can see that the prime factorization of Q looks something like this: 
Therefore, we know that 7 is the only prime factor of Q. However, this gives us no information about R so Statement (2) is not sufficient to answer the question.
Finally, let’s analyze both statements taken together:
From Statement (1), we know that P has 5 as one of its prime factors. Since 5 is a factor of P and since P is a factor of PS, then by definition, 5 is a factor of PS.
Recall that the question told us that 
. From this, we can deduce that PS must have the same factors as QR. Since 5 is a factor of PS, 5 must also be a factor of QR.
From Statement (2), we know that 7 is the only prime factor of Q. Therefore, we know that 5 is NOT a factor of Q. However, we know that 5 must be a factor of QR. The only way this can be the case is if 5 is a factor of R.
Thus, by combining both statements we can answer the question: Is R divisible by 5? Yes, it must be divisible by 5. Since BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient, the correct answer is C.
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