BTGmoderatorDC wrote:If x and y are positive integers, is xy a multiple of 8?
(1) The greatest common divisor of x and y is 10
(2) The least common multiple of x and y is 100
Source: GMAT Prep
$$x,y\,\,\, \ge \,\,1\,\,\,{\rm{ints}}$$
$${{xy} \over 8}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} $$
$$\left( 1 \right)\,\,GCD\left( {x,y} \right) = 10\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {10,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right)\,\, = \left( {10,10 \cdot 2} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,LCM\left( {x,y} \right) = 100\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,100} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right)\,\, = \left( {100,100} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,xy = GCD\left( {x,y} \right) \cdot LCM\left( {x,y} \right) = 10 \cdot 100\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
