$x_{1},...,x_{9} \in \left\{0,1,...,9\right\}$ are the first nine places of an ISBN and $p$ is the check digit, given by these digits which fulfills the condition:
$\left(\sum_{k=1}^{9}(11-k) \cdot x_{k}+p\right) \text{ mod } 11=0$
Say how it's possible to realize the following error by the deviation of the check digit: The change of one of the nine digits to any other digit without changing the remaining $8$ digits (single mistake).
This is a task of an exam from 2015 and I'd like to know how it could be solved?
The first line seems a bit confusing to me because it's saying $x_{1},...$ and after we see that $0$ is included, so why there is no $x_{0}$? But the sam starts at $1$ so I conclude that there is no $x_{0}$ And how is the check digit made up of these 0-9 digits? Is it ONE of these digits or I understand it wrong? Cannot do the task because cannot understand it :/
$x_1,\ldots x_9$ are the nine positions of the ISBM, the index (subscript) refers to the position, not its value. Each of these is a variable, which assumes any of ten possible values. For example $220056789$ corresponds to $x_1=2, x_2=2, x_3=0, \ldots$.
If we replace one digit (say, $x_1$ by $x_1'$), then the sum will change by subtracting $(11-1)\cdot x_1$, and adding $(11-1)\cdot x_1'$. Hence the sum will change by $(11-1)(x_1'-x_1)$. Note that $11-1$ is nonzero, and $x_1'-x_1$ is nonzero. Hence their product is nonzero mod $11$. Thus the sum will change by a nonzero number. This is how we detect the error; the sum is no longer zero mod $11$.