Problem Solving

BTGmoderatorDC 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
Source: Princeton Review

$$y \ge 1\,\,{\mathop{\rm int}} $$
$$x\,\,{\mathop{\rm int}} \,\,\,,\,\,\,{y \over x} = {\mathop{\rm int}} \,\,\,\left( * \right)$$
$$?\,\,\,:\,\,\,{\rm{positive}}\,\,\left( {{\rm{always}}} \right)$$

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

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.

BTGmoderatorDC 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

Strategy:
Try to show that four of the answer choices do NOT have to be positive.

Case 1: x=1 and y=1
In this case, A, B and D are not positive.
Eliminate A, B and D.

Case 2: x=1 and y=2
In this case, C is not positive.
Eliminate C.

The correct answer is E.