Find the remainder when 4⁹⁶ is divided by 6.

Answer:- 1

Explanation:-

Solution:

\begin{aligned} \frac{4^{96}}{6}, we can write it in this form \\ \frac{{(6 - 2)}^{96}}{6} \\ Now, Remainder will depend only the powers of - 2 . So, \\ \frac{{(- 2)}^{96}}{6}, it is same as \\ \frac{{({[- 2]}^{4})}^{24}}{6}, it is same as \\ \frac{{(16)}^{24}}{6} \\ Now, \\ \frac{(16 \times 16 \times 16 \times 16 . . . . . . 24 times)}{6} \\ On dividing individually 16 we always get a remainder 4 . \\ So, \\ \frac{(4 \times 4 \times 4 \times 4 . . . . . . 24 times)}{6} \\ Hence, Required Remainder = 4 \end{aligned}

$\eqalign{ & \frac{{{4^{96}}}}{6},\,{\text{we}}\,{\text{can}}\,{\text{write}}\,{\text{it}}\,{\text{in}}\,{\text{this}}\,{\text{form}} \cr & \frac{{{{\left( {6 - 2} \right)}^{96}}}}{6} \cr & {\text{Now,}}\,{\text{Remainder}}\,{\text{will}}\,{\text{depend}}\,{\text{only}}\,{\text{the}}\,{\text{powers}}\,{\text{of}}\,{\text{ - 2}}{\text{.}}\,{\text{So}}, \cr & \frac{{{{\left( { - 2} \right)}^{96}}}}{6},\,{\text{it}}\,{\text{is}}\,{\text{same}}\,{\text{as}} \cr & \frac{{{{\left( {{{\left[ { - 2} \right]}^4}} \right)}^{24}}}}{6},\,{\text{it}}\,{\text{is}}\,{\text{same}}\,{\text{as}} \cr & \frac{{{{\left( {16} \right)}^{24}}}}{6} \cr & {\text{Now}}, \cr & \frac{{\left( {16 \times 16 \times 16 \times 16\,......\,24\,{\text{times}}} \right)}}{6} \cr & {\text{On}}\,{\text{dividing}}\,{\text{individually}}\,{\text{16}}\,{\text{we}}\,{\text{always}}\,{\text{get}}\,{\text{a}}\,{\text{remainder}}\,{\text{4}}. \cr & {\text{So}}, \cr & \frac{{\left( {4 \times 4 \times 4 \times 4\,......\,24\,{\text{times}}} \right)}}{6} \cr & {\text{Hence,}}\,{\text{Required}}\,{\text{Remainder}} = 4 \cr}$
NOTE: When 4 has even number of powers, it will always give remainder 4 on dividing by 6.