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23 Aug 2021, 08:00
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E
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Difficulty:
65%
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Question Stats:
55% (02:03) correct
45%
(01:53)
wrong
based on 201
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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
(1) The highest common factor of b and d is odd
(2) The least common multiple of e and d is odd
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23 Aug 2021, 08:49
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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
(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
23 Aug 2021, 09:15
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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
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
