SajjadAhmad wrote:

If x is a factor of positive integer y, then which of the following must be positive?

A. x – y

B. y – x

C. 2x – y

D. x − 2y

E. y – x + 1

\(y \ge 1\,\,{\mathop{\rm int}}\)

\(x\,\,{\mathop{\rm int}} \,\,\,,\,\,\,{y \over x} = {\mathop{\rm int}} \,\,\,\left( * \right)\)

\(?\,\,\,:\,\,\,{\rm{positive}}\,\,\left( {{\rm{always}}} \right)\)

\(\left( {\rm{A}} \right)\,\,\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{B}} \right)\,\,\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{C}} \right)\,\,\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{D}} \right)\,\,\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

The correct answer is (E), by exclusion.

POST-MORTEM:

\(\left( {\rm{E}} \right)\,\,\,\left\{ \matrix{

\,x < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y - x + 1\,\,\, > \,\,\,0 \hfill \cr

\,x > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{y \over x} = {\mathop{\rm int}} \,\, \ge 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,y \ge x\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y - x + 1\,\,\, > 0 \hfill \cr} \right.\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)

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