Question Number 207320 by BaliramKumar last updated on 11/May/24
Commented by A5T last updated on 11/May/24
$$\mathrm{19}.\:{The}\:{remainder}\:{when}\:{a}\:{number}\:{is}\:{divided}\:{by}\:\mathrm{16} \\ $$$${is}\:{the}\:{same}\:{as}\:{the}\:{remainder}\:{when}\:{its}\:{last} \\ $$$${four}\:{digits}\:{are}\:{divided}\:{by}\:\mathrm{16} \\ $$$$\mathrm{9100}\equiv\mathrm{12}\left({mod}\:\mathrm{16}\right)\:\Rightarrow\left({b}\right) \\ $$
Answered by A5T last updated on 11/May/24
$$\mathrm{18}.\:\:\:\:\mathrm{6}^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{6};\mathrm{11}^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{1}; \\ $$$$\mathrm{9}^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{1}\:{when}\:{n}=\mathrm{2}{k};\mathrm{9}^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{9}\:{when}\:{n}=\mathrm{2}{k}+\mathrm{1} \\ $$$$\mathrm{4}^{{n}} \overset{\mathrm{10}} {\equiv}\left(−\mathrm{6}\right)^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{6}\:{when}\:{n}=\mathrm{2}{k};\:\mathrm{4}^{{n}} \overset{\mathrm{10}} {\equiv}\mathrm{4}\:{when}\:{n}=\mathrm{2}{k}+\mathrm{1} \\ $$$${when}\:{n}=\mathrm{2}{k}:\:\mathrm{4}^{{n}} +\mathrm{6}^{{n}} +\mathrm{9}^{{n}} +\mathrm{11}^{{n}} \equiv\mathrm{6}+\mathrm{6}+\mathrm{1}+\mathrm{1}\overset{\mathrm{10}} {\equiv}\mathrm{4} \\ $$$${n}=\mathrm{2}{k}+\mathrm{1}:\:\mathrm{4}^{{n}} +\mathrm{6}^{{n}} +\mathrm{9}^{{n}} +\mathrm{11}^{{n}} \equiv\mathrm{4}+\mathrm{6}+\mathrm{9}+\mathrm{1}\overset{\mathrm{10}} {\equiv}\mathrm{0} \\ $$$$\Rightarrow{Sum}\:{of}\:{distinct}\:{remainders}=\mathrm{0}+\mathrm{4}=\mathrm{4}\Rightarrow\left({b}\right) \\ $$