If P, Q, R, and S are positive integers, and P/Q = R/S, is R

First of all 140/49 does not equal to 2.87 it equals to 20/7 (it'll be a recurring decimal). So R/S=20/7 --> R is a multiple of 20 so it's a multiple of 5 too (note that we are told that all variables are positive integers).

(P / Q) * S = R

Stmt 1 says P = 140 * X

(140 * X / Q) * S = R

We do not know anything about Q. So Stmt 1 is insuff.

Stmt 2 says Q can be 7, 49 ... But we do not know anythng abt P. So Stmt 2 is insuff.

Combining both, we know R is +ve integer. So, Q has to be 7 and R is multiple of 5. C

Q can also be 35..so then the 5 would also be reduced. So how do you know that R is then still a multiple of 5? what am I missing?

Q cannot be 35 or any other multiple of 5, since it equals to \(7^{positive \ integer}\).

Hope it's clear.

Bunuel sorry, I don't quite get it

So we have P,Q,R,S are positive integers and we're trying to figure out whether R = PS/Q is divisible by 5 or if PS / 5Q is an integer right?

So Statement 1

P is divisible by 130, but I don't know nothing about the other two only that they are integers

Not sufficients

Statement 2

Q= 7^X

Clearly Insuff

Both together

I have the quesiton: is 140S / 5Q an integer where Q = 7^X

Well x could be anything and hence not sufficient

E

But I'm pretty sure there's something I'm missing, would anybody please clarify what's wrong with this line of reasoning?

Cheers
J

Yeah, the concept here is quite basic but we often overlook it.

Think about it: Is \(3^5 * 7^6 * 11^3\) divisible by 13?
The answer is simply 'No'.

For the numerator to be divisible by the denominator, the denominator MUST BE a factor of the numerator.
3^5 is only five 3s.
7^6 is only six 7s.
11^3 is only three 11s.
In the entire numerator, there is no 13 so the numerator is not divisible by 13.

On the other hand, is \(3^5 * 7^6 * 11^3 * 13\) divisible by 13? Yes, it is. 13 gets cancelled and the quotient will be \(3^5 * 7^6 * 11^3\).

Is 2^X divisible by 3? No. No matter what X is, you will only have X number of 2s in the numerator and will never have a 3. So this will not be divisible by 3.

On the same lines, in this question,

Given that \(R = \frac{(140a)*S}{7^X}\) where R is an integer.
\(R = \frac{2^2 * 5*7*a*S}{7^X}\)


So whatever X is, 7^X will get cancelled out by the numerator and we will be left with something. That something will include 5 since only 7s will be cancelled out from the numerator. Hence R is divisible by 5.

Responding to a pm:

"The second statement says that R is multiple of 5 for any x . So why are we combining the two statements ? Could you please help."

From the second statement alone, all we know is that Q is a power of 7. We have no idea about what R will be. Statement 1 tells us that P = 140 i.e. a multiple of 5. Hence we know that P must have 5 as a factor. Hence R will have a factor of 5 too.

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.

Did this by plugging values:

1) P/140 = no remainder. So P/Q = no remainder
And then we get PS = RQ --> so PS = R (140) (Lets say that Q is 140). Either way, we don't know anything about R. NS

2) Q =7^x --> cn be 7 to power 1, 2,3..least would be 7 to power 1.
P/7 = R/ S --> PS = RQ, again R is what? NS

1) + 2) --> 140 / 7 (coz P's least value is likely to be 140 and Q's least value is 7).

140/7 = R/S
S(140) = R(7)
S (4x5) = R (cancelling 7 on both sides)Therefore R is divisible by 5.
Ans C

Kudos please, if you found this useful!

if we rewrite the equation to be P * S = R * Q

1) P is divisible by 140. P is a multiple of 140

140 * S = R * Q. Let S=1, let R = 140, and let Q = 1 then yes divisible. Let S = 1, let R = 1 and let Q = 140. No not divisible.

Insufficient

2) it provides information on Q only which is not enough to decide.

Combine both now

140 * S = R * \(7^x\)

20 * 7 * S = R * \(7^x\)

P can be 140,280,420 etc.. the 7 will always have a power of 1.

R is divisible by 5 sufficient.

C

R= PS/Q
From first statement
P is divisible by 140 but we don't know about Q whether it has 5 in it or not. So insufficient
From second statement Q does not have 5 in it but we don't know about P. So insufficient
Combining we are sure that R is a multiple of 5
Hence answer is option C

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