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Which is odd?

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Which is odd?

by erjamit » Tue Jul 29, 2008 6:56 am
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. a/2

B. b/2

C. (a+b)/2

D. (a+2)/2

E. (b+2)/2

OA D

https://www.urch.com/forums/gmat-problem ... 16odd.html

I am not sure whether choices are correct since all choices seem to be wrong to me..

Thanks
Amit
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Re: Which is odd?

by Ian Stewart » Tue Jul 29, 2008 7:41 am
erjamit wrote: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. a/2

B. b/2

C. (a+b)/2

D. (a+2)/2

E. (b+2)/2
You can see that A, B, C and E can all be even by trying a = 8, b = 4 for A, B and C, and a = 8, b = 2 for E. So by elimination, D must be correct. We can prove this:

If a/b is even, then a must be even. If a is even and a - b is even, then b must also be even. So a, b and a/b are all even; a/b = x, or a = bx, where b and x are both even. So a is the product of two even numbers; a must be divisible by 4. Now look at answer D:

(a+2)/2 = (a/2) + 1

We know a/2 is even, since a is a multiple of 4. So a/2 + 1 is even + odd = odd.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by erjamit » Tue Jul 29, 2008 8:54 am
Thanks Ian. It makes sense.
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