Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If a and b are integers, is a-b an even number?
1) a^2b^2 is an even number
2) a^2+2b^2 is an even number
$$a,b\,\,{\rm{ints}}$$
$$a - b\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\mathop { \Leftrightarrow \,}\limits^{\left( * \right)} \,\,\,\,\,\boxed{\,?\,\,\,:\,\,\,\left( {a,b} \right) = \left( {{\text{odd}}\,{\text{,}}\,{\text{odd}}} \right)\,\,\,{\text{or}}\,\,\,\,\left( {a,b} \right) = \left( {{\text{even}}\,{\text{,}}\,{\text{even}}} \right)\,\,}$$
$$\left( 1 \right)\,\,{\left( {ab} \right)^2}\,\,{\rm{even}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,{a^2} + 2{b^2}\,\,{\rm{even}}\,\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,$$
$$ \Rightarrow \,\,\,\left( {\rm{E}} \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
