# ISBN Code - Construction, check digit example

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?

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You may need to drop a word why you are allowed to get from $nx\equiv 0\pmod{11}$ to $x\equiv 0\pmod{11}$. – Hagen von Eitzen Jan 14 '13 at 22:11
While you're at improving the formatting, you might want to fix the double-digit subscripts (you need to group them with braces), the asterisks (the command for a multiplication dot is \cdot) and the mod operators, for which there are various commands (\mod, \bmod, \pmod) to choose from. (Also the inline variable names aren't italicized.) – joriki Jan 18 '13 at 9:29