The remainder when (12¹³ + 23¹³) is divided by 11.

$$\eqalign{ & \frac{{{{12}^{13}}}}{{11}}\,{\text{gives}}\,{\text{Remainder}}\,1 \cr & {\text{It}}\,{\text{can}}\,{\text{be}}\,{\text{written}}\,{\text{as}}\, \cr & \frac{{\left( {12 \times 12 \times 12 \times 12\,........\,13\,\text{times}} \right)}}{{11}} \cr & {\text{On}}\,{\text{dividing}}\,{\text{it}}\,{\text{gives}}\,{\text{remainder}}\,{\text{1,}}\,{\text{each}}\,{\text{time}}. \cr & 1 \times 1 \times 1 \times 1\,.........\,13\,{\text{times}} \cr & {\text{So,}}\,{\text{final}}\,{\text{remainder}}\,{\text{will}}\,{\text{be}}\,1 \cr & \frac{{{{23}^{13}}}}{{11}} \Rightarrow \,{\text{Remainder}}\,1 \cr & {\text{Thus}}, \cr & {\text{The}}\,{\text{remainder}}\,{\text{of}}\,\frac{{\left( {{{12}^{13}} + {{23}^{13}}} \right)}}{{11}} \cr & = \left( {1 + 1} \right) \cr & = 2 \cr} $$