DivyaD wrote:If sqrt{x} and sqrt{y} are nonzero integers, what is the remainder when x is divided by y?
(1) $$\sqrt{x}\ -\ \sqrt{y} = 1$$
(2) x-y is a positive integer.

$$\left\{ \matrix{
\,x = {M^2}\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,{\mathop{\rm int}} \hfill \cr
\,y = {N^2}\,,\,\,N\mathop \ge \limits^{\left( * \right)} 1\,\,{\rm{int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\left( * \right)\,\,{\rm{WLOG}}\,\,\,\left( {{\rm{without}}\,\,{\rm{loss}}\,\,{\rm{of}}\,\,{\rm{generality}}} \right)$$

$$?\,\,\,\,:\,\,\,\,x\,\,{\rm{over}}\,\,y\,\,\,\,{\rm{remainder}}$$

$$\left( 1 \right)\,\,1 = \sqrt x - \sqrt y = M - N$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {M,N} \right) = \left( {2,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {4,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0 \hfill \cr
\,{\rm{Take}}\,\,\left( {M,N} \right) = \left( {3,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {9,4} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{INSUFF}}.$$

$$\left( 2 \right)\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,1} \right)\,\,\,\,\,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {9,4} \right)\,\,\,\,\,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.