BTGmoderatorDC wrote:If x and y are positive integers, is x^2*y^2 even ?
(1) x + 5 is a prime number
(2) y + 1 is a prime number
Source: Magoosh
$$x,y\,\, \geqslant 1\,\,\,{\text{ints}}\,\,\,\,\left( * \right)$$
$${\left( {xy} \right)^2}\,\,\mathop = \limits^? \,\,\,{\text{even}}\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,?\,\,\,:\,\,x\,\,{\text{even}}\,\,\,{\text{or}}\,\,\,y\,\,{\text{even}}\,\,}$$
$$\left( 1 \right)\,\,\left\{ \matrix{
x + 5\,\,\,\,\mathop \ge \limits^{\left( * \right)} \,\,\,6 \hfill \cr
x + 5\,\,{\rm{prime}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x + 5\,\, = {\rm{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,\,y + 1\,\,{\rm{prime}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
