BTGmoderatorLU wrote:Source: Manhattan Prep

If x and y are positive integers, what is the remainder when x^y is divided by 10?

1. x = 26
2. y^x = 1

$$x,y\,\,\, \ge 1\,\,\,{\rm{ints}}\,\,\left( * \right)$$
$$?\,\, = \,\,\left\langle {\,{x^{\,y}}} \right\rangle \,\,{\rm{ = }}\,\,{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,\,{x^{\,y}}$$
$$\left( 1 \right)\,\,x = 26\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,? = \left\langle {\,{{26}^{\,y}}} \right\rangle = 6\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\,\left[ {\,\left\langle {\,{6^{\,y}}} \right\rangle = 6\,\,,\,\,\,\forall \,\,y\,\, \ge 1\,\,{\mathop{\rm int}} \,} \right]\,\,\,\,\,\,\,\,\,$$
$$\left( 2 \right)\,\,{y^x} = 1\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,\,{\rm{ = }}\,\,\,\left\langle 1 \right\rangle \,\,\, = \,\,\,1 \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,\,{\rm{ = }}\,\,\,\left\langle 2 \right\rangle \,\,\, = \,\,\,2\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.