Chapter 2.4, Problem 32MS

ISBN-13. The 13-digit book identification number, called the International Standard Book Number (ISBN-13), has its last digit as the check digit. The check digit works on a mod 10 clock. If the ISBN-13 has digits $d_1{d}_2{d}_3{d}_4{d}_5{d}_6{d}_7{d}_8{d}_9{d}_{10},d_{11},d_{12},d_{13},$ then to check if this number is valid, we compute the following number:

$1d_1+3d_2,+1d_3+3d_4+1d_5+3d_6+1d_7,+3d_8+1d_9+3d_{10},+1d_{11}+3d_{12}+1d_{13}$

If the ISBN-13 is correct, this new calculated number should be equivalent to 0 mod 10, that is, the sum should have its last digit being 0. For example, consider the ISBN 978-0-691-15666-8. To check this ISBN, we compute (mod 10):

$\begin{array}{l}\left(1\times 9\right)+\left(3\times 7\right)+\left(1\times 8\right)+\left(3\times 0\right)+\left(1\times 6\right)+\hfill \\ \left(3\times 9\right)+\left(1\times 1\right)+\left(3\times 1\right)+\left(1\times 5\right)+\left(3\times 6\right)+\left(1\times 6\right)+\left(3\times 6\right)+\left(1\times 8\right)=\hfill \\ 9+21+8+0+6+27+1+3+5+18+6+18+8\equiv \hfill \\ 9+1+8+0+6+7+1+3+5+8+6+8+8\equiv \hfill \\ 70\equiv 0\text{ mod }10.\hfill \end{array}$

Therefore, this number is a valid ISBN-13. Verify this check method for the ISBN-13 on the copyright page of this book.