patrannn wrote:

If k and t are integer and k^2 - t^2 is an odd integer, which of the following must be an even integer ?

i)k+t+2

ii)k^2=2kt+t^2

iii)k^2+t^2

1)None

2)i only

3)ii only

4)iii only

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

This means:

(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

Eliminate "ii"

In "iii", we need to work out the solution further...

Knowing that

(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.

This means

k^2 + t^2 = even^2 + odd^2 = odd

Eliminate "iii"