Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If x and y are integers, is x^2-y^2 an even integer?
1) x^3-y^3 is an even integer
2) x+y is an even integer
BEAUTIFUL problem, Max. Congrats!
$$x,y\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$
$$\left( {x + y} \right)\left( {x - y} \right) = {x^{\text{2}}} - {y^2}\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\boxed{\,\,x + y\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,x - y\mathop = \limits^? \,\,{\text{even}}\,\,}$$
$$\left( {**} \right)\,\,\,\left\{ \matrix{
\,x + y = {\rm{even}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x - y = x + y - 2y = {\rm{even}} \hfill \cr
\,x - y = {\rm{even}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x + y = x - y + 2y = {\rm{even}} \hfill \cr} \right.$$
$$\left( 1 \right)\,\,\,{x^3} - {y^3} = {\rm{even}}\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\left\{ \matrix{
\,x\,\,{\rm{even}}\,\,,y\,\,{\rm{even}} \hfill \cr
\,{\rm{OR}} \hfill \cr
\,x\,\,{\rm{odd}}\,\,,y\,\,{\rm{odd}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x + y = {\rm{even}}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,x + y = {\rm{even}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
