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

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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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.
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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.
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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

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[odd]-[odd]t)=[odd][odd]s
s=even, p=odd,even;
s=odd, p=odd

(2) insufic

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

Ans (C)

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

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.

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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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

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