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Divisibility Problem

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Source: — Data Sufficiency |
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by Night reader » Thu Mar 17, 2011 7:48 pm
P,Q,R,S (integers) >0, Is R/5 TRUE?
PS=QR, R=PS/Q
st(1) P/140 This is Not Sufficient, we don't know Q. If Q=1 then yes, BUT if Q=140 then ?
st(2) Q=7^x, where x (integer) >0 This is Not Sufficient, as we don't know P and S what if P=1 ans S=7 ? OR P=5 and S=7?

Combined st(1&2): This is Sufficient as we know from st(2) that Q=7^x and 140/7^x is always even number for R to be integer! x can be only 1

IOM C

RadiumBall wrote:If P, Q, R, and S are positive integers, and P/Q=R/S , is R divisible by 5 ?

(1) P is divisible by 140
(2) Q = 7^x , where x is a positive integer
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by RadiumBall » Thu Mar 17, 2011 9:02 pm
Perfect OA: C
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by MAAJ » Fri Mar 18, 2011 3:09 pm
If P, Q, R, and S are positive integers, and P/Q=R/S , is R divisible by 5 ?

(1) P is divisible by 140
Means that P has 2,2,5,7...? in its prime factors (not sufficient)

(2) Q = 7^x , where x is a positive integer
Means that Q could be 7, or have infinite 7 in its prime factors (not sufficient)

(1) and (2) Combined:
P/Q=R/S
PS=RQ
P{2,2,5,7...?}*S = R*Q{7,7,7,..etc?}
Because PS has a 5 in its prime factors, then RQ must have a 5 too, but we know that Q only has 7, so R must have a 5 in its prime factors ( and also two 2) thus its divisible by 5 :mrgreen:

[spoiler]Correct Answer (C)[/spoiler]
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by madhur sharma » Fri May 29, 2015 11:21 pm
please explain it again as i did not getting properly
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by John@GMATPrepNow » Sat May 30, 2015 10:05 am
RadiumBall wrote:
If P, Q, R, and S are positive integers, and P/Q=R/S , is R divisible by 5 ?

(1) P is divisible by 140
(2) Q = 7^x , where x is a positive integer

We can rephrase by rewriting the formula as PS = QR, and writing the question as R = 5K where K is an integer.

(1) If P is divisible by 140, then we know that P = 2*2*5*7. Consequently PS = (2*2*5*7)S. Because PS is divisible by 5 and PS = RQ, then RQ must be divisible by 5 (remember, P, Q, S, and R are integers), but this doesn't mean that R is divisible by 5.

YES Example: P = 140, S = 1, Q = 140, and R = 1. R is not divisible by 5.
NO Example: P = 140, S = 1, Q = 1, and R = 140. R is divisible by 5.

Insufficient

(2) Because Q is a power of 7, QR must be divisible by 7, but since we don't know anything about PS we can't find the value of R. Again we can create a YES example and a NO example.

YES Example: P = 7, S = 1, Q = 7, and R = 1. R is not divisible by 5.
NO Example: P = 7, S = 5, Q = 7, and R = 5. R is divisible by 5.

Insufficient

(1) & (2) Now we have PS = (2*2*5*7)(S) = (7^x)(R) = QR, we know that PS and QR are both divisible by 5. But, since Q is a power of 7, it cannot be divisible by 5. Consequently, in order for QR to be divisible by 5, R must be divisible by 5.

Sufficient.

The correct answer is C

Examples and counter examples are easy ways to evaluate the individual statements, but some more abstract reasoning is required to evaluate the statements together. In the final step it's helpful to have already factored 140 and made some conclusions about PS and QR.

As a general rule, you should factor largish integers when you see them in DS problems.

J
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by Ian Stewart » Sat May 30, 2015 11:52 am
RadiumBall wrote:If P, Q, R, and S are positive integers, and P/Q=R/S , is R divisible by 5 ?

(1) P is divisible by 140
(2) Q = 7^x , where x is a positive integer
We know PS = QR. This means that PS and QR are exactly the same number. In particular, PS and QR must have exactly the same divisors.

We only care about dividing by 5. Statement 2 is clearly not sufficient alone. From Statement 1, we learn that PS is divisible by 5, so QR must be. But we can't tell if it's Q that's divisible by 5, or if it's R that is divisible by 5. But if we also use Statement 2, we know that Q is not divisible by 5, so R must be, and the answer is C.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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