GMAT CLUB OLYMPICS: If a, b, c , d, e, and f are positive integers and

Question

If a, b, c , d, e, and f are positive integers and  \frac{a}{b} - \frac{c}{d} - \frac{e}{f}=0 , is f an even integer ?

(1) The highest common factor of b and d is odd
(2) The least common multiple of e and d is odd

Kinshook · Answer

Asked: If a, b, c , d, e, and f are positive integers and  \frac{a}{b} - \frac{c}{d} - \frac{e}{f}=0 , is f an even integer ?

(1) The highest common factor of b and d is odd
If a/b = 7/9 ; c/d = 1/6; a/b - c/d = (14-1)/18 = 13/18; e/f = 13/18; f is an even integer
But If a/b = 4/9 ; c/d = 1/3; a/b - c/d = (4-3)/9 = 1/9 ; e/f = 1/9 ; f is an odd integer
NOT SUFFICIENT

(2) The least common multiple of e and d is odd
e and d both are odd
If c/d = 2/3 ; e/f = 5/6; a/b = 2/3 + 5/6 = 9/6; LCM(3,5) = 15; f is an even integer
If c/d = 2/3 ; e/f = 5/9; a/b = 2/3 + 5/9 = 11/9; LCM(3,5) = 15; f is an odd integer
NOT SUFFICIENT

(1) + (2) 
(1) The highest common factor of b and d is odd
(2) The least common multiple of e and d is odd
e and d both are odd
If a/b = 7/9 ; c/d = 2/3; a/b - c/d = 1/9 = e/f ; LCM(9,1) = 9; HCF(9,3) = 3; f=9 is an odd integer
If c/d = 2/3 ; e/f = 5/6; a/b = 2/3 + 5/6 = 9/6; LCM(3,5) = 15; HCF(6,3) = 3; f=6 is an even integer
NOT SUFFICIENT

IMO E

SChatrg · Answer

S1: means b and d can be both odd or either one is even or odd. Not suff.

S2: means both e and d is odd (if one of them was even, then LCM has to be even - prime factors) Not suff.

Both: d is odd but b can be odd / even

case 1: b is even

a / even - c / odd = odd / f

even / (odd * a - even * c) = f / odd

even / (odd * a - even) = f

even = f * a * odd - even

even = f * a * odd

even = f * a

so either a is even or f is even.

hence Option E

Lenel · Answer

If  a, b, c , d, e,  and  f  are positive integers and  \frac{a}{b}−\frac{c}{d}−\frac{e}{f}=0 , is  f  an even integer ?

(1)  The highest common factor of  b  and  d  is odd 
The highest common factor of  b  and  d  is odd implies that either  b  and  d  is odd or one of  b  and  d  is even
For example highest common factor of  3  and  5  is  1 (odd) , in this case,  \frac{a}{3}−\frac{c}{5}−\frac{e}{f}=0 
 \frac{5a-3c}{15}=\frac{e}{f}  Here,  f  is odd
For example highest common factor of  6  and  3  is  3 (odd) , in this case,  \frac{a}{6}−\frac{c}{3}−\frac{e}{f}=0 
 \frac{a-2c}{6}=\frac{e}{f}  Here,  f  is even

-Not Sufficient

(2)  The least common multiple of  e  and  d  is odd 
The least common multiple of  e  and  d  is odd implies that both  e  and  d  is odd 
For example least common multiple of  3  and  5  is  15 (odd) , in this case,  \frac{a}{b}−\frac{c}{5}−\frac{3}{f}=0 
 \frac{5a-bc}{5b}=\frac{3}{f}  Here,  f  is odd
Here, if  a=1 ,  b=2  and  c=1  then  f=10  (even)
But if  a=1 ,  b=1  and  c=2  then  f=5  (odd)

-Not Sufficient

Combining (1) and (2)  b=even/odd, d=odd, e=odd 
when  b=even, d=odd  and  e=odd 
 \frac{a}{even}−\frac{c}{odd}=\frac{odd}{f} ,  f=even 
when  b=odd, d=odd  and  e=odd 
 \frac{a}{odd}−\frac{c}{odd}=\frac{odd}{f} ,  f=odd

Answer : E

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GMAT CLUB OLYMPICS: If a, b, c , d, e, and f are positive integers and

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