I am doing a homework example and would like to ask your opinion on the correctness of the example:
How is the ISBN 3-519-42227-1 constructed, at what point is the check digit and how to calculate it. Provide a general solution, that the ISBN code will detect all single errors.
ISBN constructed: $$x_1*1 + x_2*2+...+ x_10+p$$ $$p \equiv x_1*1 + x_2*2+...+ x_10 mod 11$$ If p = 10 then p = X.
The check digit is at the last point($x_10$).
Proof that the ISBN code will detect all single errors:
Lets assume that you have 2 different ISBN Codes, which are different at point n: $$ISBN Code 1: ....a.....$$ $$ISBN Code 2: ....b.....$$ Hypothesis: a should be different to b.
$$S + n*a \equiv S + n*b mod 11$$
$$n(a-b) \equiv 0 mod 11$$ $$a-b \equiv 0 mod 11$$ $$a \equiv b mod 11 $$ because a and b have the same remainder $a=b$. If $a=b$ than this is false to our hypothesis. Therefore the codes are the same. However, that means that if these 2 codes are the same than the check digit has to be different.
My question is:
Is this solution ok? What would you approve?