Question

$\mathrm{Find}\mathrm{the}\mathrm{remainder}\mathrm{when}{2}^{34}\mathrm{is}\mathrm{divided}\mathrm{by}7.$$\left[3\right]$

Answer 1

$2^{34}\equiv x\left(\mathrm{mod}7\right)\Rightarrow {\left(2^3\right)}^{11}\times 2^1\equiv x\left(\mathrm{mod}7\right)\Rightarrow {\left(8\right)}^{11}\times 2^1\equiv x\left(\mathrm{mod}7\right)\Rightarrow {\left(1\right)}^{11}\times 2^1\left(\mathrm{mod}7\right)\equiv x\left(\mathrm{mod}7\right)[\mathrm{As},8\equiv 1(\mathrm{mod}7\left)\right]$
$\left[2\right]$
$\Rightarrow 1\times 2\left(\mathrm{mod}7\right)\equiv x\left(\mathrm{mod}7\right)\Rightarrow 2\left(\mathrm{mod}7\right)\equiv x\left(\mathrm{mod}7\right)\Rightarrow x=2\mathrm{Therefore},\mathrm{the}\mathrm{remainder}\mathrm{is}2.$$\left[1\right]$