Problem Solving

Please explain. THanks.

A quick way to do this one is to plug in some numbers.

Let's say that k=4 and t=3. These numbers are valid since 4^2 - 3^2 is an odd number.

Now, we want to prove that the answer choices COULD BE false. If they could be false even once, they must not always be even.

If you plug 4 and 3 into the answer choices, you'll find that they all come out odd. Therefore, none of them must always be even.

If you want the "theoretical" approach, one of k or t must be odd and the other must be even (i.e., they can't be the same), or else k^2 - t^2 won't come out odd. An even number squared is going to be even, and an odd number squared is going to be odd. Knowing these facts, you can see that the answer choices could be odd (but I like plugging in better).

Ans -

K and T are integers

K+t)(k-t) is odd
i.e k+t is odd and k-t is odd

1. k+t to be odd k must be odd and t must be even or vicebersa

Hence
1. K+t+2 is odd

2. sqr(k+t) is odd

3. k^2 + t^2
k is odd and t is even, above exp is odd
k even and t odd, obve exp is odd


Hence None is the even integer