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Question Number 8359 by Nayon last updated on 09/Oct/16

prove that 10 divides 11^(10) −1

$${prove}\:{that}\: \\ $$$$\mathrm{10}\:{divides}\:\mathrm{11}^{\mathrm{10}} −\mathrm{1} \\ $$

Commented by Rasheed Soomro last updated on 09/Oct/16

Actually 10 divides 11^n −1 for n≥0 11^n −1=(10+1)^n −1 All the terms of the expansion of (10+1)^n contain 10 as a factor except the last term 1. So all the terms of expansion of ′(10+1)^n −1′ contain 10 as a factor. Or 10 ∣ (11^n −1).

$$\mathrm{Actually}\:\mathrm{10}\:\mathrm{divides}\:\:\mathrm{11}^{\mathrm{n}} −\mathrm{1}\:\mathrm{for}\:\mathrm{n}\geqslant\mathrm{0} \\ $$$$\mathrm{11}^{\mathrm{n}} −\mathrm{1}=\left(\mathrm{10}+\mathrm{1}\right)^{\mathrm{n}} −\mathrm{1} \\ $$$$\mathrm{All}\:\mathrm{the}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{10}+\mathrm{1}\right)^{\mathrm{n}} \\ $$$$\mathrm{contain}\:\mathrm{10}\:\mathrm{as}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{except}\:\mathrm{the}\:\mathrm{last}\:\mathrm{term}\:\mathrm{1}. \\ $$$$\mathrm{So}\:\mathrm{all}\:\:\mathrm{the}\:\mathrm{terms}\:\mathrm{of}\:\:\mathrm{expansion}\:\mathrm{of}\:'\left(\mathrm{10}+\mathrm{1}\right)^{\mathrm{n}} −\mathrm{1}'\: \\ $$$$\mathrm{contain}\:\mathrm{10}\:\mathrm{as}\:\mathrm{a}\:\mathrm{factor}. \\ $$$$\mathrm{Or}\:\mathrm{10}\:\mid\:\left(\mathrm{11}^{\mathrm{n}} −\mathrm{1}\right). \\ $$

Answered by Rasheed Soomro last updated on 12/Oct/16

10 divides 11^(10) −1 −−−−−−−−−−−−−−−−−−−−− 11^(10) −1=(10+1)^(10) −1 = (((10)),(( 0)) )(10)^(10) + (((10)),(( 1)) )(10)^9 +...+ (((10)),(( 9)) )(10)+ (((10)),((10)) )(10)^0 −1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(10)^0 }+ (((10)),((10)) )(10)^0 −1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(10)^0 }+1−1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(1)} 10 is a factor of 11^(10) −1 Or 10 ∣ (11^(10) −1)

$$\mathrm{10}\:{divides}\:\mathrm{11}^{\mathrm{10}} −\mathrm{1} \\ $$$$−−−−−−−−−−−−−−−−−−−−− \\ $$$$\mathrm{11}^{\mathrm{10}} −\mathrm{1}=\left(\mathrm{10}+\mathrm{1}\right)^{\mathrm{10}} −\mathrm{1} \\ $$$$\:\:\:\:\:=\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{0}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{10}} +\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{1}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{9}} +...+\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{9}}\end{pmatrix}\left(\mathrm{10}\right)+\begin{pmatrix}{\mathrm{10}}\\{\mathrm{10}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{0}} −\mathrm{1} \\ $$$$\:\:\:\:\:=\left(\mathrm{10}\right)\left\{\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{0}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{9}} +\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{1}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{8}} +...+\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{9}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{0}} \right\}+\begin{pmatrix}{\mathrm{10}}\\{\mathrm{10}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{0}} −\mathrm{1} \\ $$$$\:\:\:\:\:=\left(\mathrm{10}\right)\left\{\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{0}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{9}} +\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{1}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{8}} +...+\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{9}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{0}} \right\}+\mathrm{1}−\mathrm{1} \\ $$$$\:\:\:\:\:=\left(\mathrm{10}\right)\left\{\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{0}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{9}} +\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{1}}\end{pmatrix}\left(\mathrm{10}\right)^{\mathrm{8}} +...+\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{9}}\end{pmatrix}\left(\mathrm{1}\right)\right\} \\ $$$$\:\:\:\:\:\mathrm{10}\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{11}^{\mathrm{10}} −\mathrm{1} \\ $$$$\mathrm{Or}\:\:\:\mathrm{10}\:\mid\:\:\left(\mathrm{11}^{\mathrm{10}} −\mathrm{1}\right) \\ $$

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