If p, q and r and integers, and pqr/4 is an odd integer, is p an even

Question

If p, q and r are integers, and pqr/4 is an odd integer, is p an even integer ?

(1) pq + r is an even integer

(2) qr + p is an odd integer

Mona2019 · Answer

Pqr/4 = n where n is odd
Pqr= 4n = even
So either one of the integer is odd , rest 2 are even 
Or
2 are odd, 1 is even

From statement 1,
Either pq and r ,both are odd or both are even
Both cannot be odd as we have pqr = 4n from the question statement
So both can be even
But it is insufficient to determine if p is even, as p can be odd and q can be even, making pq even.

From Statement 2,
Either of the expression is odd and the other is even,
So it is insufficient .

Combining both,
Pq is even
R is even
Statement 2's combination of Qr- odd, p - even is not possible now, 
So we are left with qr even and p odd.
So both are sufficient , C

Posted from my mobile device

GMATGuruNY · Answer

Statement 1: pq + r is an even integer 
pq=ODD and r=ODD is not viable, since pqr is a multiple of 4 and thus must be EVEN.
Implication:
pq=EVEN and r=EVEN

Case 1: p=1, q=2 and r=2, with the result that  \frac{pqr}{4} = 1  and  pq+r = 4 
In this case, the answer to the question stem is NO.
Case 2: p=2, q=1 and r=2, with the result that  \frac{pqr}{4} = 1  and  pq+r = 4 
In this case, the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statement 2: qr + p is an odd integer 
Either  qr=EVEN AND p=ODD  or  qr=ODD and p=EVEN .

The blue combination is satisfied by the values in Case 1 above:
Case 1: q=2, r=2 and p=1, with the result that  \frac{pqr}{4} = 1  and  qr+p = 5 
In this case, the answer to the question stem is NO.
An example of the red combination:
Case 3: q=1, r=1 and p=4, with the result that  \frac{pqr}{4} = 1  and  qr+p = 5 
In this case, the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 3, INSUFFICIENT.

Statements combined: 
The red combination is not viable, since Statement 1 requires that r=EVEN.
Since only the blue combination is viable, p=ODD.
Thus, the answer to the question stem is NO.
SUFFICIENT.

C

Archit3110 · Answer

let pqr/4 = 9 ; pqr = 36
36= (1,3,12) ; ( 2,3,6) ( p,q,r)
target is p odd or even
#1
pq + r is an even integer
possible when 
2*3+6 or 3*2+6
p can be odd or even insufficient
#2
qr + p is an odd integer
Possible when
1*3+12 ; 2*6+3 ; p can be odd or even
insufficient
from 1 &2
value of p has to be odd pair set ( 3,2,6) ( p,q,r)
option C

SaquibHGMATWhiz · Answer

Solution:      
  Pre Analysis:  
  p, q and r are integers  
 pqr/4 is an odd integer which means   pqr = 4	imes odd   
 We are asked if p is an even integer or not

Statement 1:   pq + r is an even integer
  pq + r will be even when  pq and r are either both odd or both even  
 However, both cannot be odd otherwise  pqr = 4	imes odd  will not be satisfied 
 This is possible both when p is even or odd   p = even (say 2), q = odd, and r = even (say 2)    p = odd, q = even (say 2), and r = even (say 2)    
 Thus,  statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2:   qr + p is an odd integer
  qr + p is odd when  qr and p are off opposite nature i.e., even-odd or odd-even   
 This is possible both when p is even or odd   p = even (say 4), q = odd, and r = odd    p = odd, q = even (say 2), and r = even (say 2)    p = odd, q = even (say 4), and r = odd   p = odd, q = odd, and r = even (say 4)      Thus,  statement 2 alone is also not sufficient

Combining:   
  Upon combining, we see that only one case where  p = odd, q = even (say 2), and r = even (say 2)  is common between the 2 statements  
 Thus, we can say that p is odd and not even

Hence the right answer is    Option C

bumpbot · Answer

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If p, q and r and integers, and pqr/4 is an odd integer, is p an even

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If p, q and r and integers, and pqr/4 is an odd integer, is p an even

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