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