$11$ divisibility 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.

HINT $\ $ Specialize $\rm\ x = 10\ $ below
$$\rm(x+1)\ (a_n\ x^n +\:\cdots\:+a_1\ x + a_0)\ =\ a_n\ x^{n+1}+ (a_n+a_{n-1})\ x^{n}+\:\cdots\:(a_1+a_0)\ x+ a_0$$
