ISBN check digit

Question

a ISBN-10 has 10 digits. The correctness of the last digit, the check digit, can be verified using the following formula:

$$(x_1 + 2x_2 + 3x_3 + 4 x_4 + 5x_5 + 6x_6 + 7x_7 + 8x_8+9x_9+10m) \text{ mod } 11$$

Why is it also possible to use the formula:

$$(10x_1 + 9x_2 + 8x_3 + 7 x_4 + 6x_5 + 5x_6 + 4x_7 + 3x_8+2x_9+m) \text{ mod } 11$$

I tried to write everyting as an equation, but I don't get it.

Can anyone please give me a hint?

Thanks.

Answer 1

Add your two formulas together, you will get

$(11 \times (x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + m)) \mod 11$

which the result must be $0$.

So, if the first formula evaluates to $0$, the second formula must also evaluate to $0$, because:

$x \mod 11 = 0$ and $(x + y) \mod 11 = 0 \Longrightarrow y \mod 11 = 0$

This explains the meaning of "the second is the additive inverse of the first for $\mod 11$ operation".

Answer 2

Modulo $11$ the second is the additive inverse of the first, and thus the first is $0$ if and only if the second is $0$.