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# If a,b and c are positive integers and only one out of them is even

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Math Expert
Joined: 02 Aug 2009
Posts: 7764
If a,b and c are positive integers and only one out of them is even  [#permalink]

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10 Jul 2018, 20:18
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55% (hard)

Question Stats:

53% (02:03) correct 47% (01:25) wrong based on 71 sessions

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If a,b and c are positive integers and only one out of them is even, which of the following must be even?

I. $$\frac{ab}{c}$$, where ab/c is an integer.
II. abc
III. a+b+c

A. Only I
B. Only II
C. Only III
D. Only II and III
E. I, II and III

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Re: If a,b and c are positive integers and only one out of them is even  [#permalink]

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10 Jul 2018, 20:42
with one of the numbers even and two of them odd, the product of abc HAS to be even.
Since Odd + Odd = Even and Even + Even = Even; the sum of abc will also be even.
If ab/c is an integer, it means that an odd denominator is cancelling an odd numerator, finally leading to an even integer.
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Re: If a,b and c are positive integers and only one out of them is even  [#permalink]

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10 Jul 2018, 21:25
1
chetan2u wrote:
If a,b and c are positive integers and only one out of them is even, which of the following must be even?

I. $$\frac{ab}{c}$$, where ab/c is an integer.
II. abc
III. a+b+c

A. Only I
B. Only II
C. Only III
D. Only II and III
E. I, II and III

New question

Given, a,b and c are positive integers & only one out of them is even.

I.$$\frac{ab}{c}$$,
Since ab/c is an integer, hence $$Rem(\frac{numerator}{denominator})=0$$, so, when one of the integers is even, then numerator/denominator must be a multiple of 2.hence the expression must be even.
II.abc
Since only one out of a,b, and c is odd; hence even*(odd)=Even. hence abc is Even.
Must be even.
III. a+b+c
when a=even, b, c=odd; a+b+c=even+odd+odd=even+even=even
Since a,b, and c are added ;same results would be obtained when b or c is even.
Must be even.
Ans. (E)
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Re: If a,b and c are positive integers and only one out of them is even  [#permalink]

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10 Jul 2018, 22:26
IMO E is the correct choice.

we have one even and 2 odd numbers.Cases I, II, III will all be True.

Thanks!

chetan2u wrote:
If a,b and c are positive integers and only one out of them is even, which of the following must be even?

I. $$\frac{ab}{c}$$, where ab/c is an integer.
II. abc
III. a+b+c

A. Only I
B. Only II
C. Only III
D. Only II and III
E. I, II and III

New question

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Re: If a,b and c are positive integers and only one out of them is even   [#permalink] 10 Jul 2018, 22:26
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