Question

Are each of these eight-digit codes possible ISSNs? That is, do they end with a correct check digit?

a) 1059-1027

b) 0002-9890

c) 1530-8669

d) 1007-120 x

Short answer

a) Thus the check digit is $\mathbf{8}$, while the last digit of the given number is $\mathbf{7}$. Since they are different, the ISSN number is not valid.

b) Thus the check digit is $\mathbf{0}$, while the last digit of the given number is $\mathbf{0}$. Since they are the same, the ISSN number is valid.

c) Thus the check digit is $\mathbf{9}$, while the last digit of the given number is $\mathbf{9}$. Since they are the same, the ISSN number is valid.

d) Thus the check digit is $\mathbf{2}$, while the last digit of the given number is $\mathbf{x}$(which corresponds with 10 ). Since they are different, the ISSN number is not valid.

Step-by-step solution

Step 1

For a International Standard Serial Number (ISSN)${\mathbf{d}}_{\mathbf{1}}{\mathbf{d}}_{\mathbf{2}}\mathbf{\dots }{\mathbf{d}}_{\mathbf{8}}$, the check digit can be calculated using:

${{\mathbf{d}}_{\mathbf{8}}}^{\mathbf{0}}\mathbf{3}{\mathbf{d}}_{\mathbf{1}}\mathbf{+}\mathbf{4}{\mathbf{d}}_{\mathbf{2}}\mathbf{+}\mathbf{5}{\mathbf{d}}_{\mathbf{3}}\mathbf{+}\mathbf{6}{\mathbf{d}}_{\mathbf{4}}\mathbf{+}\mathbf{7}{\mathbf{d}}_{\mathbf{5}}\mathbf{+}\mathbf{8}{\mathbf{d}}_{\mathbf{6}}\mathbf{+}\mathbf{9}{\mathbf{d}}_{\mathbf{7}}\mathbf{(}\mathbf{mod}\mathbf{11}\mathbf{)}$

Note: When ${{\mathbf{d}}_{\mathbf{8}}}^{\mathbf{0}}\mathbf{10}\mathbf{(}\mathbf{mod}\mathbf{11}\mathbf{)}\mathbf{,}\mathbf{x}$is used to represents ${\mathbf{d}}_{\mathbf{-}}\mathbf{\left\{}\mathbf{8}\mathbf{\right\}}$

Step2

(a) Given :

$\mathbf{1059}\mathbf{-}\mathbf{1027}$

Use the above expression to calculate the check digit:

$$\begin{gathered} {\mathbf{d}}_{\mathbf{8}}\mathbf{=}\mathbf{3}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{\left(}\mathbf{9}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{8}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{9}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{3}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{25}\mathbf{+}\mathbf{54}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{18}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{107}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{8} \end{gathered}$$

Thus the check digit is 8 , while the last digit of the given number is 7 . Since they are different, the ISSN number is not valid

Step 3

(b) Given :

$\mathbf{0002}\mathbf{-}\mathbf{9890}$

Use the above expression to calculate the check digit:

$$\begin{gathered} {\mathit{d}}_{\mathbf{8}}\mathbf{=}\mathbf{3}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{\left(}\mathbf{9}\mathbf{\right)}\mathbf{+}\mathbf{8}\mathbf{\left(}\mathbf{8}\mathbf{\right)}\mathbf{+}\mathbf{9}\mathbf{\left(}\mathbf{9}\mathbf{\right)}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{63}\mathbf{+}\mathbf{64}\mathbf{+}\mathbf{81}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{220}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{0} \end{gathered}$$

Thus the check digit is 0 , while the last digit of the given number is 0 . Since they are the same, the ISSN number is valid.

Step 4

(c) Given :

$\mathbf{1530}\mathbf{-}\mathbf{8669}$

Use the above expression to calculate the check digit:

$$\begin{gathered} {\mathit{d}}_{\mathbf{8}}\mathbf{=}\mathbf{3}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{\left(}\mathbf{8}\mathbf{\right)}\mathbf{+}\mathbf{8}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{+}\mathbf{9}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{3}\mathbf{+}\mathbf{20}\mathbf{+}\mathbf{15}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{56}\mathbf{+}\mathbf{48}\mathbf{+}\mathbf{54}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{196}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{9} \end{gathered}$$

Thus the check digit is 9, while the last digit of the given number is 9 . Since they are the same, the ISSN number is valid.

Step 5

(d) Given :

$\mathbf{1007}\mathbf{-}\mathbf{120}\mathbf{x}$

Use the above expression to calculate the check digit:

$$\begin{gathered} {\mathbf{d}}_{\mathbf{8}}\mathbf{=}\mathbf{3}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{\left(}\mathbf{7}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{8}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{+}\mathbf{9}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{3}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{42}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{16}\mathbf{+}\mathbf{0}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{68}\mathbf{mod}\mathbf{11} \\ \mathbf{=}\mathbf{2} \end{gathered}$$

Thus the check digit is 2 , while the last digit of the given number is X (which corresponds with 10 ). Since they are different, the ISSN number is not valid.