AbhishekRyu wrote:is x/140 an integer?
1) LCM of x & y is 360
2) GCF of x & 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 therefore (A).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
