Bunuel wrote:
Is \(\frac{a^2 - b^2}{2}\) an odd integer?
(1) \(a-b\) is odd
(2) \(a^2 + b^2\) is odd
Note that the result of \(\frac{a^2 - b^2}{2} = \frac{(a - b)(a + b)}{2}\) might not be an integer, if the result is not an integer then it is not an odd integer. Then for this particular question, the harder answer to find is the "yes" case and we should focus on trying to find such a case. One more thing to note is we don't even know if a and b are integers.
Statement 1:It would be helpful to know something on \(a + b\). If we want an odd result, we should let \(a + b\) be a multiple of 2, but not a multiple of 4.
\(a + b\) being a multiple of 4 would cause the result to be even (and we'd answer "no" which is already the easier case to find).
We can try \((a, b) = (1.5, 0.5)\), which gives \(a + b = 2\) and \(a - b = 1\). Then the result is 1 which is odd, so insufficient.
Statement 2:Again let us try to find the case where the result is odd.
We can let \(a^2 + b^2 = 3\) and to get \(a^2 - b^2\) we have to subtract \(2b^2\), so let \(2b^2 = 1\) then we would have \(a^2 - b^2 = 2\), giving us an odd result. (a's value can be deducted but there's no point in finding it). Thus insufficient.
Combined:From statement 1 we know \(a^2 - 2ab + b^2\) is odd, and since \(a^2 + b^2\) is odd we know \(2ab\) has to be even, and \(ab\) must be an integer. Combine that with \(a - b\) must be odd, we cannot have values such as 3.5 - 0.5 anymore so \(a\) and \(b\) must both be integers.
Given they are both integers we must have 1 even 1 odd, then the result will be a fraction so it is not even an integer. Sufficient.
Ans: C