rxs0005 wrote:
If P, Q, R, and S are positive integers, and \(\frac{P}{Q} = \frac{R}{S}\), is R divisible by 5?
(1) P is divisible by 140
(2)\(Q = 7^x\), where x is a positive integer
OFFICIAL SOLUTION
Let's begin by analyzing the information given to us in the question:
If P, Q, R, and S are positive integers, and \(\frac{P}{Q} = \frac{R}{S}\), 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:
\(\frac{P}{Q} = \frac{R}{S}\) → \(PS=RQ\)
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: \(140=2*2*5*7\).
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: \(Q = 7^x\), where x is a positive integer.
From this, we can see that the prime factorization of Q looks something like this: \(Q=7*7*7......\) 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 \(PS=RQ\). 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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