Question Number 204717 by BaliramKumar last updated on 26/Feb/24
![((3^0 +3^1 +3^2 +.........+ 3^(200) )/(13)) =^(Remainder) ?](https://www.tinkutara.com/question/Q204717.png)
$$\frac{\mathrm{3}^{\mathrm{0}} +\mathrm{3}^{\mathrm{1}} +\mathrm{3}^{\mathrm{2}} \:+………+\:\mathrm{3}^{\mathrm{200}} }{\mathrm{13}}\:\overset{\mathrm{Remainder}} {=}\:? \\ $$
Answered by Rasheed.Sindhi last updated on 26/Feb/24
![3^0 +3^1 +3^2 +...+3^(200) ≡(1+3+9)×67(mod 13) ≡13×67≡0(mod 13](https://www.tinkutara.com/question/Q204718.png)
$$\mathrm{3}^{\mathrm{0}} +\mathrm{3}^{\mathrm{1}} +\mathrm{3}^{\mathrm{2}} +…+\mathrm{3}^{\mathrm{200}} \equiv\left(\mathrm{1}+\mathrm{3}+\mathrm{9}\right)×\mathrm{67}\left({mod}\:\mathrm{13}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\mathrm{13}×\mathrm{67}\equiv\mathrm{0}\left({mod}\:\mathrm{13}\right. \\ $$
Commented by BaliramKumar last updated on 26/Feb/24
![thanks](https://www.tinkutara.com/question/Q204719.png)
$$\mathrm{thanks} \\ $$