Although the system has recently changed, most older books have been assigned a unique 10 -digit number called an International Standard Book or ISBN . The first 9 digits are split into three groups - the first group representing the country or group of countries in which the book originates, the second identifying the publisher, and the third assigned to the book title itself. The tenth and final digit, called a check digit, is computed from the first nine digits and is used to ensure that an electronic transmission of the ISBN, say over the Internet, occurs without error. To explain how this is done, regard the first nine digits of the ISBN as a vector $\mathbf{b}$ in $R^{9}$, and let a be the vector $$ \mathbf{a}=(1,2,3,4,5,6,7,8,9) $$ Then the check digit $c$ is computed using the following procedure: 1. Form the dot product $\mathbf{a} \cdot \mathbf{b}$. 2. Divide $\mathbf{a} \cdot \mathbf{b}$ by 11 , thereby producing a remainder $c$ that is an integer between 0 and 10 , inclusive. The check digit is taken to be $c$, with the proviso that $c=10$ is written as $\mathrm{X}$ to avoid double digits. For example, the ISBN of the brief edition of Calculus, sixth edition, by Howard Anton is $$ 0-471-15307-9 $$ which has a check digit of $9 .$ This is consistent with the first nine digits of the ISBN, since $$ \mathbf{a} \cdot \mathbf{b}=(1,2,3,4,5,6,7,8,9) \cdot(0,4,7,1,1,5,3,0,7)=152 $$ Dividing 152 by 11 produces a quotient of 13 and a remainder of 9 , so the check digit is $c=9 .$ If an electronic order is placed for a book with a certain ISBN, then the warehouse can use the above procedure to verify that the check digit is consistent with the first nine digits, thereby reducing the possibility of a costly shipping error.