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Question Number 192893 by mnjuly1970 last updated on 30/May/23

$$$$$$\mathrm{Q}:\mathrm{Find}\mathrm{the}\mathrm{remainder}\mathrm{of}\mathrm{dividing}$$$$\mathrm{the}\mathrm{following}\mathrm{number}\mathrm{by}7.$$$$$$$$\mathrm{N}=3^{{10}^1}+3^{{10}^2}+3^{{10}^3}+...+3^{{10}^{10}}$$$$$$$$...@\mathrm{nice}-\mathrm{mathematics}...$$$$$$$$$$

Answered by leodera last updated on 30/May/23

$${10}^n\equiv 0mod\left(2\right)$$$${10}^n\equiv 1mod\left(3\right)$$$$$$$$usingC.R.T$$$${10}^n=4mod\left(6\right)$$$$$$$$a^{\mathrm{\Phi }\left(7\right)}\equiv a^6\equiv 1mod\left(7\right)$$$$$$$$3^{{10}^n}\equiv 3^{\left({10}^{n}mod\left(6\right)\right)}\equiv 3^4\equiv 4mod\left(7\right)$$$$$$$$3^{{10}^1}+{10}^{{10}^2}+......+3^{{10}^{10}}\equiv \underbrace{4+4+..\underset{10times}{.}.......+4}mod\left(7\right)$$$$\equiv 4\times 10\equiv 5mod7$$$$$$$$theremainderis5$$$$$$$$$$$$$$

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