Data Sufficiency

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 is D. Please explain with steps.

For a-b to be even, either both are even or both are odd

For a/b to be even, a has to be even

To satisfy both conditions, a and b both are even

Since, both are even none of the condtion given in question can be odd...

Anyone else????

We have to go a bit further. Even integer is an integer that is divisible by 2. A multiple of 4 is also an even integer but with the exception that the number when divided by 2 is again even number.

a - b is even means that either both a and b are even or both are odd.
a/b is even not only means that both are even, but also means that a has to be a multiple of 4. One can verify with some numbers.

The key here is understand the last point that a is multiple of 4.
(a + 2)/2 = a/2 + 1.
And as we know that a/2 is even a/2 + 1 must be odd...

Question is asking for a condition that is always true (MUST).

a/2+1 will be odd only when a is multiple of 4

If you do the same logic for b, for that matter, E can be true also.

B has to be an even factor of A... in other words, A has to be a multiple of both 2 and another even number (or simply, divisible by 4). This isn't the case for B... it can be any even number.

If A is divisible by 4 and has B as a factor, A/2 +1 would always be odd.

Again, since B doesn't have to be divisible by 4, (B+2)/2 won't always be even.

Got it...