k^2 - t^2 is odd. what must be odd:
I) k+t+2
II) k^2 + 2kt + t^2
III) k^2 + t^2
There are a few ways to look at this. First, when dealing with integers, notice that addition and subtraction follow the same rules where evens and odds are concerned. That is,
if a+b is even, then a-b is even
if a+b is odd, then a-b is odd
So if k^2 - t^2 is odd, so is k^2+t^2. So III) is definitely odd. From here you can see that II) must also be odd, but I think it's more illuminating if you factor: k^2 - t^2 is a difference of squares, so
k^2 - t^2 = (k+t)(k-t)
We know (k+t)(k-t) is odd, and the only way you can have an odd product of integers is if those integers are odd. Thus k+t and k-t are odd. From there, I) must be odd, and II) must be as well, since it's the same as (k+t)^2. This question is partly testing whether you recognize two of the most common 'factoring patterns': a^2 - b^2 and a^2 + 2ab + b^2.
