P, Q, R , S, T, U are positive integers, such that S/P + T/Q = U/R, is

Question

Competition Mode Question

P, Q, R , S, T, U are positive integers, such that  \frac{S}{P} + \frac{T}{Q} = \frac{U}{R} , is P odd?

(1) Q and R are odd
(2) S is odd

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freedom128 · Answer

S/P = U/R - T/Q

(1) Q and R are odd  
Let U=1,R=1,T=2,Q=3. Thus, S/P = 1/1 - 2/3 = 1/3. 
---> S/P can be either 1/3 (P is odd) or 2/6 (P is even). We don't know which one should be selected. 
NOT SUFFICIENT

(2) S is odd 
Case A: Let U=1,R=1,T=2,Q=3. Thus, S/P = 1/1 - 2/3 = 1/3. ---> If S is odd (ie. 1,3,5), P must be odd (ie. 3,9,15).
Case B: Let U=1,R=1,T=1,Q=2. Thus, S/P = 1/1 - 1/2 = 1/2. ---> If S is odd (ie. 1,3,5), P must be even (ie. 2,6,10).
We don't know which case should be selected.
NOT SUFFICIENT

(1)+(2) 
Now we know that only case A in (2) is applicable. Let U=1,R=1,T=2,Q=3. Thus, S/P = 1/1 - 2/3 = 1/3. ---> If S is odd (ie. 1,3,5), P must be odd (ie. 3,9,15).
Testing also with U=2,R=1,T=1,Q=3,S=odd and another case, I confidently conclude that P must be odd. 
SUFFICIENT

FINAL ANSWER IS (C) 
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unraveled · Answer

P, Q, R , S, T, U are positive integers, such that  \frac{S}{P} + \frac{T}{Q} = \frac{U}{R} , is P odd?

(1) Q and R are odd
 \frac{S}{P} + \frac{T}{o} = \frac{U}{o} 
Case I: P ODD
 \frac{e}{o} + \frac{o}{o} = \frac{o}{o} 
 \frac{6}{3} + \frac{1}{1} = \frac{9}{3}

Case II: P EVEN
 \frac{e}{e} + \frac{o}{o} = \frac{e}{o} 
 \frac{10}{2} + \frac{9}{3} = \frac{24}{3}

INSUFFICIENT.

(2) S is odd
Many possibilities exist.
Case I: P ODD
 \frac{9}{3} + \frac{6}{3} = \frac{15}{3}

Case II: P EVEN
 \frac{9}{2} + \frac{9}{2} = \frac{18}{2}

INSUFFICIENT.

Together 1 and 2
 \frac{S}{P} = \frac{U}{R} - \frac{T}{Q} 
 \frac{S}{P} = \frac{U*Q - R*T}{R*Q} 
S*R*Q = P*(U*Q - R*T)
o*o*o = o*(o*o - o*e) OR 
o*o*o = o*(e*o - o*o)
So, neither P nor (U*Q - R*T) can be even

Case I: P ODD
 \frac{9}{3} + \frac{6}{3} = \frac{25}{5}

Case II: P EVEN 
  \frac{9}{2} + \frac{4}{2} = \frac{13}{2}

SUFFICIENT.
Answer C.

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • P, Q, R, S, and T are positive integers.
•  \frac{S}{P} + \frac{T}{Q} =\frac{U}{R} 
 o  P = \frac{RQS}{(QU-TR)}   
We need to find whether P is Odd or not.
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  Q and R are odd.
 •  P= \frac{Odd*Odd*S}{(Odd*U-T*Odd)} 
 o Since we don't have any information about S, U, and T, we can't answer the question.  
Hence, statement 1 is not sufficient, we can eliminate answer options A and D.
 Statement 2:  S is odd
 •  P = \frac{RQ*Odd}{(QU-TR)} 
 o Since we don't have any information about Q, R, S, U, and T, we can't answer the question.  
Hence, statement 2 is also not sufficient, we can eliminate answer options B.
 Step 3: Analyse Statements by combining. 
From statement 1: Q and R are odd
From statement 2: S is odd
 •  P = \frac{Odd*Odd*Odd}{(QU-TR)} = \frac{Odd}{(QU-TR)} 
 o It is given that P is a positive integer
o Irrespecive of even and odd nature of U and T, P will be an odd number
  This is because, we know that  \frac{Odd}{even}  is not an integer and  \frac{odd}{odd}  is always odd.   
Hence, the correct answer is  Option C.

monikakumar · Answer

P, Q, R , S, T, U are positive integers, such that S/P+T/Q=U/R, is P odd?

(1) Q and R are odd
(2) S is odd
1)
s/p=u/r-t/q
s/p=qu-tr/rq => rq->odd
p is odd

2)
no relationship

Ans A

exc4libur · Answer

s/p+t/q=u/r
sq+tp=upq/r
rsq+rtp=upq
upq-rtp=rsq
p(uq-rt)=rsq

(1) insufic
p(uq-rt)=rsq
p(u - t)=  s
s=even, p=odd,even;
s=odd, p=odd

(2) insufic

(1/2) sufic
p(uq-rt)=rsq
p(u - t)=   
p(u - t)=odd
only an odd*odd=odd
p=odd

Ans (C)

madgmat2019 · Answer

P, Q, R , S, T, U are positive integers, such that S/P+T/Q=U/R.................. P odd?

(1) Q and R are odd

eg: Q=3,P=had to be odd to make R to be odd
if P is going to be even, then R becomes even........ SUFFICIENT

(2) S is odd

Value of S is not going to be helpful in finding the value of P.........INSUFFICIENT

OA:A

minustark · Answer

S/P + T/Q = U/R
1) Q and R are odd.
S/P = U/R - T/Q or, S/P = (QU -TR)/RQ or, P = SRQ/(QU-TR)......(i). Depending on the value of S, T and U, P may be odd or even. Lets say, P is odd. In that case both of the numerator and denominator have to be either Odd or Even.
1st case : Both are odd: S is odd. Between U and T, one is even and another is odd.
2nd Case: Both are even. S is even. Q and T are both odd or even. 
Two options possible, not sufficient.
2) S is odd. From question stem, we get the equation (i). Now the value of P is dependent on the values of the other four integers. Clearly, not sufficient.
Together, SRQ is odd. Since P is an integer, it has to be an odd integer. Cause if (QU -TR) is even, P will be a fraction. Sufficient.
C is the answer.

yashikaaggarwal · Answer

Answer is supposedly A 
S/P+T/Q=U/R
Statement 1: Q and R are odd
S/P=U/R-T/Q
Since R are Q are odd.
There LCM will always be odd.
Giving P as odd.
(Sufficient)

Statement 2: S is Odd. No other constraint is given to determine P. (Not sufficient)

So IOC is A

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Crytiocanalyst · Answer

P, Q, R , S, T, U are positive integers, such that  \frac{S}{P} + \frac{T}{Q} = \frac{U}{R} , is P odd?

(1) Q and R are odd
P is odd for S=odd 
P is even for S= even 
Clearly insufficient

(2) S is odd
 P= even for Q=even and R = odd 
 P= odd for Q=odd and R =odd

However when 1 and 2 is combined P has only one possibility P = odd

Therefore IMO C

bumpbot · Answer

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P, Q, R , S, T, U are positive integers, such that S/P + T/Q = U/R, is

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P, Q, R , S, T, U are positive integers, such that S/P + T/Q = U/R, is

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