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Question Number 56845 by Tawa1 last updated on 25/Mar/19

Answered by math1967 last updated on 25/Mar/19

(11...108times)×10+1+(22..108times)×10+2 ......(77..108times)×10+7 ((11....108times+.....77...108times)/(37))+((1+..7)/(37)) remainder 0 +remainder28 so 28 ans

$$\left(\mathrm{11}...\mathrm{108}{times}\right)×\mathrm{10}+\mathrm{1}+\left(\mathrm{22}..\mathrm{108}{times}\right)×\mathrm{10}+\mathrm{2} \\ $$$$......\left(\mathrm{77}..\mathrm{108}{times}\right)×\mathrm{10}+\mathrm{7} \\ $$$$\frac{\mathrm{11}....\mathrm{108}{times}+.....\mathrm{77}...\mathrm{108}{times}}{\mathrm{37}}+\frac{\mathrm{1}+..\mathrm{7}}{\mathrm{37}} \\ $$$${remainder}\:\mathrm{0}\:+{remainder}\mathrm{28} \\ $$$${so}\:\mathrm{28}\:{ans} \\ $$

Commented by Tawa1 last updated on 25/Mar/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Answered by mr W last updated on 25/Mar/19

=1×111...11+2×111...11+3×111...11+...+7×111...11 =(1+2+3+...+7)×111...11 =28×111...11_(109 times) =280×111...11_(108 times) +28 =280×(111×10^(105) +111×10^(102) +...+111×10^3 +111)+28 =280×(37×3×10^(105) +37×3×10^(102) +...+37×3×10^3 +111)+28 =280×37×3(10^(105) +10^(102) +...+10^3 +1)+28 =37×N+28 ⇒remainder=28

$$=\mathrm{1}×\mathrm{111}...\mathrm{11}+\mathrm{2}×\mathrm{111}...\mathrm{11}+\mathrm{3}×\mathrm{111}...\mathrm{11}+...+\mathrm{7}×\mathrm{111}...\mathrm{11} \\ $$$$=\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{7}\right)×\mathrm{111}...\mathrm{11} \\ $$$$=\mathrm{28}×\underset{\mathrm{109}\:{times}} {\mathrm{111}...\mathrm{11}} \\ $$$$=\mathrm{280}×\underset{\mathrm{108}\:{times}} {\mathrm{111}...\mathrm{11}}+\mathrm{28} \\ $$$$=\mathrm{280}×\left(\mathrm{111}×\mathrm{10}^{\mathrm{105}} +\mathrm{111}×\mathrm{10}^{\mathrm{102}} +...+\mathrm{111}×\mathrm{10}^{\mathrm{3}} +\mathrm{111}\right)+\mathrm{28} \\ $$$$=\mathrm{280}×\left(\mathrm{37}×\mathrm{3}×\mathrm{10}^{\mathrm{105}} +\mathrm{37}×\mathrm{3}×\mathrm{10}^{\mathrm{102}} +...+\mathrm{37}×\mathrm{3}×\mathrm{10}^{\mathrm{3}} +\mathrm{111}\right)+\mathrm{28} \\ $$$$=\mathrm{280}×\mathrm{37}×\mathrm{3}\left(\mathrm{10}^{\mathrm{105}} +\mathrm{10}^{\mathrm{102}} +...+\mathrm{10}^{\mathrm{3}} +\mathrm{1}\right)+\mathrm{28} \\ $$$$=\mathrm{37}×{N}+\mathrm{28} \\ $$$$\Rightarrow{remainder}=\mathrm{28} \\ $$

Commented by mr W last updated on 25/Mar/19

generally: all digits 3n times ⇒ remainder 0 all digits 3n+1 times ⇒ remainder 28 all digits 3n+2 times ⇒ remainder 12

$${generally}: \\ $$$${all}\:{digits}\:\mathrm{3}{n}\:{times}\:\Rightarrow\:{remainder}\:\mathrm{0} \\ $$$${all}\:{digits}\:\mathrm{3}{n}+\mathrm{1}\:{times}\:\Rightarrow\:{remainder}\:\mathrm{28} \\ $$$${all}\:{digits}\:\mathrm{3}{n}+\mathrm{2}\:{times}\:\Rightarrow\:{remainder}\:\mathrm{12} \\ $$

Commented by Tawa1 last updated on 25/Mar/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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