If positive integers x and y are not both odd, which of the

Even +/- odd= odd
Even +/- even= even
Odd +/- odd = even

Even * even= even
Even * odd= even
Odd * odd= odd

X & Y could be both even or one odd and one even=> x*y= even always.
Ans.A

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I am sorry but i disagree as x and y are not both odd means that either one of them is odd i.e. x or y is odd which implies the one that is not odd is even. so basically x is odd and y is even or x is even and y is odd.

Nope. x and y are not both odd means that:

(i) Both x and y are even;
OR:
(ii) One of x or y is even, the other odd.

Given: integers x and y are not both odd

There are two different cases that satisfy this information:
Case i: Both numbers are even
Case ii: One number is even and one number is odd

Let's examine each case individually...

Case i: Both numbers are even
So it could be the case that x = 2 and y = 2, so let's plug these values into our answer choices

(A) (2)(2) = 4, which is even. Keep.
(B) 2 + 2 = 4, which is even. Keep.
(C) 2 - 2 = 0, which is even. Keep.
(D) 2 + 2 - 1 = 3, which is odd. ELIMINATE.
(E) 2(2 + 2) - 1 = 7, which is odd. ELIMINATE.

We're down to A, B and C

Case ii: One number is even and one number is odd
So it could be the case that x = 1 and y = 2, so let's plug these values into the remaining answer choices
(A) (1)(2) = 2, which is even. Keep.
(B) 1 + 2 = 3, which is odd. ELIMINATE.
(C) 1 - 2 = -1, which is odd. ELIMINATE.

By the process of elimination, the correct answer is A

Cheers,
Brent

If both x and y are both even, then shouldn't x + y always equal even (as even + even = even)? In this case, B could be the answer.

I think the question meant that ATLEAST ONE, i.e either x or y, must be odd. In that case, A is a definitive answer.

Thoughts?

I think this is explained in the solution you quote: Positive integers x and y are NOT both odd, means that either both x and y are even or one is even and the other one is odd.

Option B is not correct because if one is even and the other one is odd, then x + y = even + odd = odd.

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