Is the product 2*x*5*y an even integer?

1. 2 + x + 5 + y is an even integer

2. x - y is an odd integer

(C) 2008 GMAT Club - m16#18

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

* EACH statement ALONE is sufficient

* Statements (1) and (2) TOGETHER are NOT sufficient

I did get the right answer, but I overshot on time. I have a fundamental question that I want to ask you guys:

if x + y = Odd and

x - y = Odd , then can we conclude that x and y will always be integers? In the question solution we did prove it algebraically and thus I am inclined to "remember" this fact rather than solve it, if encountered again in a problem.

Extrapolating from the official explanation:

x+y=n ....1

x-y=m.....2

1: x=n-y , subst in 2

2 can be re-written as 2y=(n-m)

LHS is always Even - this implies y is an Integer

Examine RHS : n-m , now difference of 2 Integers is Even when both are odd or both are even. Thus we can conclude the following:

if x+y = Odd and x-y = Odd then both x, and y are integers.

if x+y = Even and x-y = Even then both x, and y are integers.

FYI - O.A. is C