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# If a and b are positive integers such that a b and a/b are

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VP
Joined: 22 Nov 2007
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If a and b are positive integers such that a b and a/b are [#permalink]

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07 Mar 2008, 12:28
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If a and b are positive integers such that a – b and a/b are both even integers,
which of the following must be an odd integer?
a/2
b/2
(a+b)/2
(a+2)/2
(b+2)/2

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Director
Joined: 10 Sep 2007
Posts: 933

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07 Mar 2008, 14:30
The ways a - b can be even is that either both a, b are even or both of them are odd.
The ways a / b can be even is that a, b are even (also considering above that both numbers are either even or odd, as odd/odd cannot result in even so both numbers should be even)

Now we know both numbers are even, so question is which of following must be odd.

a/2 No as a can be multiple of 4 as well.
b/2 No as b can be multiple of 4 as well.
(a+b)/2 No as both a and b can be multiple of 4 as well.
(a+2)/2 = a/2 + 1 As a is even number a/2 may or may not be even.
(b+2)/2 = b/2 + 1 As b is even number b/2 may or may not be even.

So there is no clear cut answer. If at all I will choose between D, & E. By picking some numbers.

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CEO
Joined: 29 Mar 2007
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07 Mar 2008, 15:03
abhijit_sen wrote:
The ways a - b can be even is that either both a, b are even or both of them are odd.
The ways a / b can be even is that a, b are even (also considering above that both numbers are either even or odd, as odd/odd cannot result in even so both numbers should be even)

Now we know both numbers are even, so question is which of following must be odd.

a/2 No as a can be multiple of 4 as well.
b/2 No as b can be multiple of 4 as well.
(a+b)/2 No as both a and b can be multiple of 4 as well.
(a+2)/2 = a/2 + 1 As a is even number a/2 may or may not be even.
(b+2)/2 = b/2 + 1 As b is even number b/2 may or may not be even.

So there is no clear cut answer. If at all I will choose between D, & E. By picking some numbers.

I agree this problem isnt clear.

Essentially simplifying for D and E we have D: A/2+1 E: b/2+1

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VP
Joined: 22 Nov 2007
Posts: 1078

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08 Mar 2008, 01:28
I agree this problem isnt clear.

Essentially simplifying for D and E we have D: A/2+1 E: b/2+1[/quote]

I think the problem is clear but quite tricky. I'll explain (source gmatprep)

we know that a-b is an even number and that a=bx, where both b and x are even.
it follows that a is a multiple of 4. any multiple of 4, if added up with 2, can't be any longer divisible by 4. thus we would have that a+2 is the product of 2 and an odd integer. OA is D

Last edited by marcodonzelli on 19 Mar 2008, 03:15, edited 1 time in total.

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Director
Joined: 05 Jan 2008
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08 Mar 2008, 05:21
I got D, I picked numbers and went ahead,

A B
32 8
24 6

48 24

(A+2)/2 is odd

but (B+2)/2 is not odd all the time

Thus D
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Re: odd even integ   [#permalink] 08 Mar 2008, 05:21
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# If a and b are positive integers such that a b and a/b are

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