DivyaD wrote:If x and y are consecutive integers such that x < y and x is odd, which of the following must be true?
I. xy - y is odd.
II. x(x + y) is even.
III. x^2 + y^2 is odd.

a) None
b) III only
c) I and II
d) I and III
e) II and III

$$\left. \matrix{
x\,\,{\rm{odd}} \hfill \cr
y = x + 1\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,?\,\,:\,\,{\rm{true}}$$

$$\left( {\rm{I}} \right)\,\,y\left( {x - 1} \right) = \underbrace {\left( {x + 1} \right)}_{{\rm{even}}}\underbrace {\left( {x - 1} \right)}_{{\rm{even}}}\,\,\mathop {\rm{ = }}\limits^? \,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{C}} \right),\left( {\rm{D}} \right)\,\,{\rm{refuted}}$$
$$\left( {{\rm{II}}} \right)\,\,x\left( {x + y} \right) = \underbrace {\,\,x\,\,}_{{\rm{odd}}}\underbrace {\left( {2x + 1} \right)}_{{\rm{odd}}}\,\,\mathop {\rm{ = }}\limits^? \,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)\,\,{\rm{refuted}}\,\,\,\left( {{\rm{also}}} \right)$$
$$\left( {{\rm{III}}} \right)\,\,{x^2} + {y^2} = \underbrace {\,\,{x^2}\,\,}_{{\rm{odd}}} + \underbrace {{{\left( {x + 1} \right)}^2}}_{{\rm{even}}}\,\,\mathop {\rm{ = }}\limits^? \,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{B}} \right)\,\,$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.