Question

The ISBN-10 of the sixth edition of Elementary Number

Theory and Its Applications is 0-321-500Q1-8, where Q

is a digit. Find the value of Q.

Short answer

The value of Q is 3

Step-by-step solution

Step 1

$\begin{array}{r}\mathbf{0}\mathbf{-}\mathbf{321}\mathbf{-}\mathbf{500}\mathbf{Q}\mathbf{1}\mathbf{-}\mathbf{8}\mathbf{}\mathbf{}\mathbf{}\\ \\ {\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{3}}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{5}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \\ {\mathbf{x}}_{\mathbf{6}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{7}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{8}}\mathbf{=}\mathbf{Q}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{9}}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ {\mathbf{x}}_{\mathbf{10}}\mathbf{=}\mathbf{8}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\end{array}$

For a ISBN-10 code ${\mathbf{x}}_1{\mathbf{x}}_2\dots {\mathbf{x}}_{10}$, the check digit is calculated using the following expression:

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

Use this expression to calculate the check digit:

localid="1668605080714" $$\begin{gathered} {\mathbf{x}}_{\mathbf{10}}\mathbf{=}\mathbf{1}\mathbf{\times }\mathbf{0}\mathbf{+}\mathbf{2}\mathbf{\times }\mathbf{3}\mathbf{+}\mathbf{3}\mathbf{\times }\mathbf{2}\mathbf{+}\mathbf{4}\mathbf{\times }\mathbf{1}\mathbf{+}\mathbf{5}\mathbf{\times }\mathbf{5}\mathbf{+}\mathbf{6}\mathbf{\times }\mathbf{0}\mathbf{+}\mathbf{7}\mathbf{\times }\mathbf{0}\mathbf{+}\mathbf{8}\mathbf{\times }\mathbf{Q}\mathbf{+}\mathbf{9}\mathbf{\times }\mathbf{1}\mathbf{mod}\mathbf{11}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{=}\mathbf{0}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{25}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{8}\mathit{Q}\mathbf{+}\mathbf{9}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{8}\mathit{Q}\mathbf{+}\mathbf{50}\mathbf{mod}\mathbf{11} \end{gathered}$$

The check digit is known to be 8.

$\mathbf{8}\mathit{Q}\mathbf{+}\mathbf{50}\mathbf{mod}\mathbf{11}\mathbf{=}\mathbf{8}$

Since$\mathbf{50}\mathbf{mod}\mathbf{11}\mathbf{=}\mathbf{8}$

Subtract 6 from each side of equation

Step 2

Since the inverse of $\mathbf{8}\mathbf{mod}\mathbf{117}\mathbf{mod}\mathbf{11}$we should multiply both sides of the equation by $\mathbf{7}$:

role="math" localid="1668604956425" $\begin{array}{r}\mathbf{7}\mathbf{\times }\mathbf{8}\mathbf{Q}\mathbf{mod}\mathbf{11}\mathbf{=}\mathbf{7}\mathbf{\times }\mathbf{2}\mathbf{mod}\mathbf{11}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{56}\mathbf{Q}\mathbf{mod}\mathbf{11}\mathbf{=}\mathbf{14}\mathbf{mod}\mathbf{11}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{\text{Qmod}}\mathbf{11}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\end{array}$

Since Q is a digit (between0 and 9 ), Q then has to be equal to 3.