Max@Math Revolution wrote:[GMAT math practice question]
If x, y are integers, is (x-y)(x+y)(x^2+y^2) an odd number?
1) x is an odd number
2) x-y is an odd number
$$x,y\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$$A\left( {x,y} \right) = \left( {x - y} \right)\left( {x + y} \right)\left( {{x^2} + {y^2}} \right)\,\,\,\mathop = \limits^? \,\,\,{\rm{odd}}$$
$$\left( 1 \right)\,\,x\,\,{\rm{odd}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,A\left( {1,0} \right) = 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\,A\left( {1,1} \right) = 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,x - y\,\,{\rm{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,x + y\,\,{\rm{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\,\,\,{x^2} + {y^2}\,\,{\rm{odd}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( {**} \right)\,\,\left\{ \matrix{
\,x + y\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,\underbrace {x - y}_{{\rm{odd}}} + \underbrace {2y}_{{\rm{even}}}\,\,\, = \,\,\,{\rm{odd}} \hfill \cr
\,{x^2} + {y^2}\,\,{\rm{even}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,x,y\,\,\,{\rm{both}}\,\,{\rm{odd}}\,\,{\rm{or}}\,\,{\rm{both}}\,\,{\rm{even}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {2} \right)} \,\,\,\,\,{\rm{impossible}} \hfill \cr} \right.$$
The correct answer is therefore (B).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
