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Combining 1 and 2
r=s=t or rst = r^3
3 of pqrst are integers , has to be r,s and as they are =
pqr^3 = 4
pq= 4/r^3
r^3 can never be 4
so p cannot be 1/q
IMO E.

if we're getting a definite NO, won't the answer be C in that case.

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Regards,
Harsha

and moreover, what is the reason behind directly combining both the statements?? why is that you negate 1 and 2??

Absolutely!

(1) If r = s = t = k, which is either positive or negative, then p q = 4/k^3, and for p = 1/q be true, p q = 1.

Two questions now,

First one being:

Is 4/k^3 equal to 1, which we don’t know or

The second one being:

Can 4/k^3 be equal to 1? Well, the answer is NO for a rational k, but yes for an irrational k.

Insufficient

(2) When three of p, q, r, s, and t are integers, many possible integers n in the range -2 < n < 2, with permissible repetitions are possible for the three of p, q, r, s, and t, leading to no fixed status of p and q.

Insufficient

Take them one now


If r = s = t = k, which is either positive or negative, and three of p, q, r, s, and t are integers, then at least one of r, s, and t is an integer, and if it’s so, then k is an integer, and if it’s really so then no integer has its cube equal to 4, hence p is definitely NOT equal to 1/q.

C

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Spot on.
It's a silly mistake on my part.

IMOC

it should be C

Is pq=1 ?

given pq * rst = 4; if rst=4 then pq=1

st(1) r=s=t, we don't know if r^3=4 Not Sufficient
st(2) r,s,t integers then pq may and may not be 1, as rst={1*1*1, 1*2*2,...} Not Sufficient

Combining st(1&2) r=s=t, when r=1 --> pq*1=4 False; when r=2 --> pq*8=4 False Sufficient

I am going with C on this on as with
1) you get p=4/qr^3. tells nothing
2)3 of them are integers. tells nothing.

But when you combine them, then you know : rst are those 3 integers that are equal.

Thus,

In order for p=1/q then r^3 =4, as they are integers. No way it is possible. Hence, I select C

Need Statement 2, otherwise we bring Fractions to the party.
Need Statement 1, to restrict all kinds of integers to the party too.

Therefore to answer we would need both.

Hence C.

Statement 1: r=s=t
the given statement is pqrst=4; pqr^3=4; pq=4/r^3
Now, as there is no integer or fraction whose cube is equal to 4 our RHS in the equation (pq=4/r^3) can never be 1. thus, pq is not equal to 1. Thus, p is not equal to 1/q. Statement 1 is sufficient

Statement 2: three of p, q, r, s & t are integers. Now, this can be any of the three integers. Thus, insufficient to point out whether pq=1 or not. To illustrate:
a) Let p=32, q=1, r=1/2, s=1/4, t=1 this would satisfy pqrst=4. However, pq is not equal to 1.
b) Let p=1/2, q=2, r=1, s=1, t=4 this would satisfy pqrst=4 and pq=1
Thus, statement 2 is not sufficient.

Thus, the answer is A

Perfect explanation, thanks Mitch.

S1 and S2 are alone not sufficient here as there is absolutely no clue to prove or disprove p = 1/q.
Combining, pq*r^3 = 4 => pq = 4/r^3. No way can this be 1. As if r,s and t are integers, pq cannot be 1.

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