$$x,y\,\, \ge 1\,\,{\rm{ints}}\,$$swerve wrote:If x and y are positive integers, what is the remainder when 10^x+y is divided by y?
1) x=50
2) y=2
Source: Official Guide
$$?\,\,\,:\,\,\,\left( {{{10}^x} + y} \right)\,\,{\rm{remainder}}\,\,{\rm{when}}\,\,{\rm{divided}}\,\,{\rm{by}}\,\,y$$
$$\left( 1 \right)\,\,x = 50\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,y = 2\,\,\,\, \Rightarrow \,\,\,\,\,{10^{50}} + 2\,\,{\rm{even}}\,\,\,\, \Rightarrow \,\,\,\,\,? = 0\, \hfill \cr
\,{\rm{Take}}\,\,y = 3\,\,\,\, \Rightarrow \,\,\,\,\,{10^{50}} + 3\,\,\,{\rm{not}}\,\,{\rm{divisible}}\,\,{\rm{by }}\,{\rm{3}}\,\,\,\,\left( {\sum {\,{\rm{digits}}\,\,\,{\rm{ = }}\,\,{\rm{4}}} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? \ne 0\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,y = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,{10^{\,x\, \ge \,1}} + 2\,\,\,{\rm{even}}\,\,\,\, \Rightarrow \,\,\,\,\,? = 0\,$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
