Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If x and y are positive integers, is $$\sqrt{15xy}$$ an integer?
1) xy is a multiple of 15
2) x and y are prime numbers
$$x,y\,\, \ge 1\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$$\sqrt {15xy} \,\,\mathop = \limits^? \,\,\operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\boxed{\,xy\,\,\mathop = \limits^? \,\,15 \cdot {M^2}\,\,\,,\,\,\,M \geqslant 1\,\,\operatorname{int} \,}$$
$$\left( 1 \right)\,\,{{xy} \over {15}} = {\mathop{\rm int}} \,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,15} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,15} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,x,y\,\,{\rm{primes}}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {3,5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\,\left( {x,y} \right) = \left( {3,5} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( {5,3} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
