Problem Solving

If both x and y are positive integers then which of the following expressions must also be even?
i.. y^x-1
ii.y-1
iii. x/2

(a) i only
(b) ii only
(c) iii. only
(d) i and iii only
(e) i, ii, and iii

I feel that some info is missing here.

Lets take (ii) and (iii) into consideration first.

(ii) will be even if y is odd and odd is y is even.

(iii) will be even if x is even and odd is x is odd.

Now considering(i)

y ^ (x-1)

Consider x as odd ( say 3) and y as even ( say 2). So after substituting in the equation we get value as 4 viz even. Now consider, x as even ( say 2) and y as odd ( say 3). Now we get value as 3 viz odd.

According to my solution none of them "must" be even.

I think that the problem states that both x and y are even positive integers. In this case the answer is (d) : (i) and (iii) only

everything's eventual wrote:I think that the problem states that both x and y are even positive integers. In this case the answer is (d) : (i) and (iii) only

(iii) is not necessarily an even. Take x=6, then x/2 would be Odd.

@ Shalabh...yes very silly of me...in that case we have only (i)