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Question Number 126666 by bramlexs22 last updated on 23/Dec/20

$$3+33+333+3333+...+\underset{2020times}{\underbrace{3333...3}}$$$$divideby2.findtheremainder$$

Answered by JDamian last updated on 23/Dec/20

$$itiseasy$$$$0$$

Answered by liberty last updated on 23/Dec/20

$$S=3(1+11+111+1111+...+\underset{2020times}{\underbrace{1111...1}})$$$$S=3(1+(1+10)+(1+10+{10}^2)+...+(1+10+{10}^2+...+{10}^{2019}))$$$$S=3(2020+\left(10\times 2019\right)+\left({10}^2\times 2018\right)+...+{10}^{2019})$$$$2\mid S=0\left(mod2\right)$$

Answered by talminator2856791 last updated on 23/Dec/20

$$$$$$\mathrm{to}\mathrm{find}\mathrm{remainder}\mathrm{of}3+33+333+.......+\underset{2020\mathrm{times}}{\underbrace{3333......3}}$$$$\mathrm{is}\mathrm{equivalent}\mathrm{finding}\mathrm{whether}\mathrm{it}\mathrm{is}\mathrm{even}\mathrm{or}\mathrm{odd}.$$$$\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{any}\mathrm{two}\mathrm{odd}\mathrm{numbers}\mathrm{is}\mathrm{even}.$$$$\mathrm{any}\mathrm{multiple}\mathrm{of}10\mathrm{is}\mathrm{even}\left(\mathrm{as}10\mathrm{is}\mathrm{even}\right)$$$$$$$$k=3+33+333+......+\underset{2020\mathrm{times}}{\underbrace{3333....3}}$$$$k=3+(30+3)+(330+3)+......+(\underset{2019\mathrm{times}}{\underbrace{3333.....3}0}+3)$$$$k=\underset{20203^{\prime }\mathrm{s}}{\underbrace{3+3+3+......+3}}+10j$$$$k=\underset{1010\mathrm{times}}{\underbrace{6+6+6+......+6}}+10j$$$$\Rightarrow k\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}\mathrm{and}\mathrm{therefore}$$$$\mathrm{remainder}\mathrm{of}0\mathrm{when}\mathrm{divided}\mathrm{by}2$$$$$$$$$$

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