Find the remainder when 67¹⁰⁷ is divided by 7.

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{{67}^{107}}}}{7} \cr & Or,\,\frac{{{{\left( {7 \times 9 + 4} \right)}^{107}}}}{7} \cr & {\text{The}}\,{\text{remainder}}\,{\text{will}}\,{\text{be}}\,{\text{same}}\,{\text{as}} \cr & \frac{{{4^{107}}}}{7} \cr & Or,\,\frac{{{4^3}}}{7} \cr & Or,\,\frac{{64}}{7} \cr & {\text{Required}}\,{\text{Remainder}} = 1 \cr} $$