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We know that $1331$ is divisible by $11$. As per the $11$ divisibility test, we can say $1331$ is divisible by $11$. However we cannot get any quotient. If we subtract each unit digit in the following way, we can see the quotient when $1331$ is divided by $11$.

$1331 \implies 133 -1= 132$
$132 \implies 13 - 2= 11$
$11 \implies 1-1 = 0$, which is divisible by $11$.

Also the quotient is, arrange, all the subtracted unit digits (in bold and italic) from bottom to top, we get $121$. Which is quotient. I want to know how this method is working? Please write a proof.

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    @Benjamin, I want to know what is wrong with saying "I want" as it doesn't seem impolite at all. Also the question seems clear enough.2011-10-11

2 Answers 2

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Look how the method works for the number $abcde$. You substract $e$ from $abcd$, getting a'b'c'd', and you apply the method to a'b'c'd'. If the method works for a'b'c'd', it yields n' such that a'b'c'd'=11\times n'. But a'b'c'd'0 is a'b'c'd'\times 10=11\times 10\times n' and abcde=a'b'c'd'\times 10+e\times 11 hence abcde=11\times 10\times n'+11\times e is 11\times(10\times n'+e). This proves that n=10\times n'+e is indeed the correct answer for $abcde$.

Edit (Upon OP's request, the same proof, with more apparatus but with zero more mathematics.)

Look how the method works for the number $N=a_ka_{k-1}\cdots a_2a_1a_0$ with $k\geqslant1$. You substract $a_0$ from $a_ka_{k-1}\cdots a_2a_1$, getting $M=b_kb_{k-1}\cdots b_2b_1$, and you apply the method to $M$. If the method works for $M$, it yields $m$ such that $M=11\times m$. But $b_kb_{k-1}\cdots b_2b_10$ is $M\times 10=11\times 10\times m$ and $N=M\times 10+a_0\times 11$ hence $N=11\times 10\times m+11\times a_0$ is $11\times(10\times m+a_0)$. This proves that $n=10\times m+a_0$ is indeed the correct answer for $N$ if $m$ was the correct answer for $M$. A recursion on the number of digits of $N$ yields the result.

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    @J.M. Recall prior meta discussion on such matters. Nevertheless, I too am curious why it was downvoted.2011-10-11
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HINT $\ $ Specialize $\rm\ x = 10\ $ below

$\rm(x+1)\ (a_n\ x^n +\:\cdots\:+a_1\ x + a_0)\ =\ a_n\ x^{n+1}+ (a_n+a_{n-1})\ x^{n}+\:\cdots\:(a_1+a_0)\ x+ a_0$

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    Thank you sir. I need little bit more e$x$planation. Could you?2011-10-11