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3-33-333-3333-3333-3-2020-times-divide-by-2-find-the-remainder-

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Question Number 126666 by bramlexs22 last updated on 23/Dec/20
3+33+333+3333+...+3333...3_(2020 times) divide by 2. find the remainder
$$\:\mathrm{3}+\mathrm{33}+\mathrm{333}+\mathrm{3333}+…+\underset{\mathrm{2020}\:{times}} {\underbrace{\mathrm{3333}…\mathrm{3}}}\: \\ $$$${divide}\:{by}\:\mathrm{2}.\:{find}\:{the}\:{remainder} \\ $$
Answered by JDamian last updated on 23/Dec/20
it is easy 0
$${it}\:{is}\:{easy} \\ $$$$\mathrm{0} \\ $$
Answered by liberty last updated on 23/Dec/20
S= 3(1+11+111+1111+...+1111...1_(2020 times) ) S = 3(1+(1+10)+(1+10+10^2 )+...+(1+10+10^2 +...+10^(2019) )) S=3(2020+(10×2019)+(10^2 ×2018)+...+10^(2019) ) 2∣S = 0 (mod 2 )
$${S}=\:\mathrm{3}\left(\mathrm{1}+\mathrm{11}+\mathrm{111}+\mathrm{1111}+…+\underset{\mathrm{2020}\:{times}} {\underbrace{\mathrm{1111}…\mathrm{1}}}\right) \\ $$$${S}\:=\:\mathrm{3}\left(\mathrm{1}+\left(\mathrm{1}+\mathrm{10}\right)+\left(\mathrm{1}+\mathrm{10}+\mathrm{10}^{\mathrm{2}} \right)+…+\left(\mathrm{1}+\mathrm{10}+\mathrm{10}^{\mathrm{2}} +…+\mathrm{10}^{\mathrm{2019}} \right)\right) \\ $$$${S}=\mathrm{3}\left(\mathrm{2020}+\left(\mathrm{10}×\mathrm{2019}\right)+\left(\mathrm{10}^{\mathrm{2}} ×\mathrm{2018}\right)+…+\mathrm{10}^{\mathrm{2019}} \right) \\ $$$$\mathrm{2}\mid{S}\:=\:\mathrm{0}\:\left({mod}\:\mathrm{2}\:\right) \\ $$
Answered by talminator2856791 last updated on 23/Dec/20
to find remainder of 3+33+333+.......+3333......3_(2020 times) is equivalent finding whether it is even or odd. the sum of any two odd numbers is even. any multiple of 10 is even (as 10 is even) k = 3+33+333+......+3333....3_(2020 times) k = 3+(30+3)+(330+3)+......+(3333.....3_(2019 times) 0+3) k = 3+3+3+......+3_(2020 3′s) + 10j k = 6+6+6+......+6_(1010 times) + 10j ⇒ k is an even number and therefore remainder of 0 when divided by 2
$$\: \\ $$$$\:\mathrm{to}\:\mathrm{find}\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{3}+\mathrm{33}+\mathrm{333}+…….+\underset{\mathrm{2020}\:\mathrm{times}} {\underbrace{\mathrm{3333}……\mathrm{3}}}\: \\ $$$$\:\mathrm{is}\:\mathrm{equivalent}\:\mathrm{finding}\:\mathrm{whether}\:\mathrm{it}\:\mathrm{is}\:\mathrm{even}\:\mathrm{or}\:\mathrm{odd}. \\ $$$$\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{any}\:\mathrm{two}\:\mathrm{odd}\:\mathrm{numbers}\:\mathrm{is}\:\mathrm{even}. \\ $$$$\:\mathrm{any}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{10}\:\mathrm{is}\:\mathrm{even}\:\left(\mathrm{as}\:\mathrm{10}\:\mathrm{is}\:\mathrm{even}\right) \\ $$$$\: \\ $$$$\:{k}\:=\:\mathrm{3}+\mathrm{33}+\mathrm{333}+……+\underset{\mathrm{2020}\:\mathrm{times}} {\underbrace{\mathrm{3333}….\mathrm{3}}}\: \\ $$$$\:{k}\:=\:\mathrm{3}+\left(\mathrm{30}+\mathrm{3}\right)+\left(\mathrm{330}+\mathrm{3}\right)+……+\left(\underset{\mathrm{2019}\:\mathrm{times}} {\underbrace{\mathrm{3333}…..\mathrm{3}}0}+\mathrm{3}\right) \\ $$$$\:{k}\:=\:\underset{\mathrm{2020}\:\mathrm{3}'\mathrm{s}} {\underbrace{\mathrm{3}+\mathrm{3}+\mathrm{3}+……+\mathrm{3}}}\:+\:\mathrm{10}{j} \\ $$$$\:{k}\:=\:\underset{\mathrm{1010}\:\mathrm{times}} {\underbrace{\mathrm{6}+\mathrm{6}+\mathrm{6}+……+\mathrm{6}}}\:+\:\mathrm{10}{j} \\ $$$$\:\Rightarrow\:{k}\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{number}\:\mathrm{and}\:\mathrm{therefore}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{0}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{2} \\ $$$$\: \\ $$$$\: \\ $$

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