Bunuel wrote:
Is x/140 an integer?
(1) The least common multiple (LCM) of x and y is 360.
(2) The greatest common factor (GCF) of x and y is 40.
\({x \over {{2^2} \cdot 5 \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}}\)
\(\left( 1 \right)\,\,\,LCM\left( {x,y} \right) = 360\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,x\,\,{\mathop{\rm int}} \,\,\,\,({\rm{implicitly}}) \hfill \cr
\,x\,\,{\rm{is}}\,{\rm{a}}\,{\rm{factor}}\,{\rm{of}}\,\,360\,\,\,\,\, \Rightarrow \,\,\,\,\,{{{2^3} \cdot {3^2} \cdot 5} \over x} = {\mathop{\rm int}} \,\,\,\left( * \right) \hfill \cr} \right.\)
\(\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,7\,\,{\rm{is}}\,\,{\rm{not}}\,\,{\rm{a}}\,\,{\rm{factor}}\,\,{\rm{of}}\,\,x\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x \over {{2^2} \cdot 5 \cdot 7}}\,\,\, \ne \,\,\,{\mathop{\rm int}} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\)
\(\left( 2 \right)\,\,\,GCF\left( {x,y} \right) = 40 = {2^3} \cdot 5\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {{2^3} \cdot 5 \cdot 7,{2^3} \cdot 5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {{2^3} \cdot 5,{2^3} \cdot 5 \cdot 7} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
The correct answer is (A), indeed.
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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