If k and t are integer and k^2 - t^2 is an odd integer, which of the following must be an even integer ?
5)i,ii and iii
The answer is None.
We know that k^2 - t^2 = odd = (k+t)*(k-t)
We also know that only
odd*odd = odd
(k+t) = odd & (k-t) = odd
Now, Eliminate "i" because odd + 2 will always be odd.
In "ii", I think this is what you mean:
k^2 - 2kt + t^2 = (k-t)*(k-t) = odd*odd = odd
In "iii", we need to work out the solution further...
(k+t) = odd
(k-t) = odd
There are two cases here:
odd + even = odd
even + odd = odd
This means k and t will never be odd&odd or even&even pair. They will always be either even&odd or odd&even. In other word, they will always be different.
k^2 + t^2 = even^2 + odd^2 = odd