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.