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Question Number 137383 by bramlexs22 last updated on 02/Apr/21

$$Whatistheremainder{13}^{163}when$$$$dividedby99$$

Answered by EDWIN88 last updated on 02/Apr/21

$$\mathrm{By}\mathrm{Euler}\mathrm{Theorem}$$$$\phi \left(99\right)=3(3-1)(11-1)=60$$$${\left({13}^{60}\right)}^2\times {\left(13\right)}^{43}={13}^{43}\left(\mathrm{mod}99\right)$$$$=85\left(\mathrm{mod}99\right)$$

Answered by physicstutes last updated on 02/Apr/21

$${13}^{60}=a\left(99\right)+1,\mathrm{where}a\in \mathbb{R}$$$$\Rightarrow {13}^{60}\equiv 1\left(\mathrm{mod}99\right)$$$$163=2\left(60\right)+43$$$$\Rightarrow {13}^{60}={13}^{2\left(60\right)}.{13}^{43}$$$$\Rightarrow {13}^{60}={13}^{2\left(60\right)}.{13}^{43}\equiv 1\times {13}^{43}\left(\mathrm{mod}99\right)$$$$\mathrm{but}{13}^{43}\equiv 85\left(\mathrm{mod}99\right)$$$$\Rightarrow {13}^{60}\equiv 85\left(\mathrm{mod}99\right)$$$$\mathrm{remainder}=85$$

Answered by Rasheed.Sindhi last updated on 02/Apr/21

$$99=3^2\times 11$$$$\varphi \left(99\right)=99(1-\frac{1}{3})(1-\frac{1}{11})$$$$=99.\frac{2}{3}.\frac{10}{11}=60$$$${13}^{60}\equiv 1\left(mod99\right)$$$$30\mid 60\wedge {13}^{30}\equiv 1\left(mod99\right)$$$${\left({13}^{30}\right)}^5\equiv {\left(1\right)}^5\left(mod99\right)$$$${13}^{150}.{13}^{13}\equiv 1.{13}^{13}\equiv 85\left(mod99\right)$$

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