Question

Explain how to check digit is found for a 10-digit ISBN

Short answer

$x_{10}\equiv x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}(mod11)$

Step-by-step solution

ISBN

ISBN (International Standard Book Number) is a 10 digit code${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{x}}_{\mathbf{10}}$assigned by publisher.

To check the digit is selected so that

${\mathit{x}}_{\mathbf{10}}\mathbf{=}\mathbf{\sum }_{\mathbf{i}\mathbf{=}\mathbf{1}}^{\mathbf{9}}\mathbf{ }\mathit{i}{\mathit{x}}_{\mathbf{i}}\mathbf{(}\mathbf{mod}\mathbf{11}\mathbf{)}$

Explaining how to check digit

For a ISBN 10 code, the check digit is calculated as

$$\begin{gathered} x_{10}\equiv \sum _{i=1}^9 i{x}_i(mod11) \\ \equiv x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}(mod11) \end{gathered}$$