Question

Determine whether each of these 13-digit numbers is a

valid.

a) $\mathbf{978}\mathbf{-}\mathbf{0}\mathbf{-}\mathbf{073}\mathbf{-}\mathbf{20679}\mathbf{-}\mathbf{1}$

b)$\mathbf{978}\mathbf{-}\mathbf{0}\mathbf{-}\mathbf{45424}\mathbf{-}\mathbf{521}\mathbf{-}\mathbf{1}$

c)$\mathbf{978}\mathbf{-}\mathbf{3}\mathbf{-}\mathbf{16}\mathbf{-}\mathbf{148410}\mathbf{-}\mathbf{0}$

d)$\mathbf{978}\mathbf{-}\mathbf{0}\mathbf{-}\mathbf{201}\mathbf{-}\mathbf{10179}\mathbf{-}\mathbf{9}$

Short answer

(a) $978-0-073-20679-1$is not a valid number.

(b) $978-0-45424-521-1$is not a valid number.

(b) $978-3-16-148410-0$is a valid number.

(c)$978-0-201-10179-9$ is a valid number.

Step-by-step solution

Identifying the data given in the question

Given in the question,

The check digit $a_{13}$for an ISBN-13 with initial digits $a_1{a}_2\dots a_{12}$ is determined by the congruence

$\left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4\dots +a_{12}\right)\equiv 0(mod10)$

(a) Step 2: Determining that 978−0−073−20679−1 is a valid number.

Given number is

$978-0-073-20679-1$

The check digit for an ISBN-13 with initial digits $a_1{a}_2\dots a_{12}$ is determined by the congruence

$\left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4\dots +a_{12}\right)\equiv 0(mod10)$

To verify substituting the values of digits

$\begin{array}{r}\equiv (9+8+0+3+0+7+1)+3(7+0+7+2+6+9)(mod10)\\ \equiv 28+3\left(31\right)(mod10)\\ \equiv 121(mod10)\\ \equiv 1\end{array}$

Since the result is 1 not zero, therefore$978-0-073-20679-1$ is not a valid number.

(b) Step 3: Determining that  978−0−45424−521−1 is a valid number.

Given number is

$978-0-45424-521-1$

The check digit $a_{13}$for an ISBN-13 with initial digits $a_1{a}_2\dots a_{12}$ is determined by the congruence

$\left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4\dots +a_{12}\right)\equiv 0(mod10)$

To verify substituting the values of digits

$\begin{array}{r}\equiv (9+8+4+4+4+2+1)+3(7+0+5+2+5+1)(mod10)\\ \equiv 32+3\left(20\right)(mod10)\\ \equiv 92(mod10)\\ \equiv 2\end{array}$

Since the result is 2 not zero, therefore $978-0-45424-521-1$is not a valid number.

(c) Step 4: Determining that 978−3−16−148410−0 is a valid number.

Given number is

$978-3-16-148410-0$

The check digit ${\mathrm{a}}_{13}$for an ISBN-13 with initial digits $a_1{a}_2\dots a_{12}$ is determined by the congruence

$\left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4\dots +a_{12}\right)\equiv 0(mod10)$

To verify substituting the values of digits

$\begin{array}{r}\equiv (9+8+1+1+8+1+0)+3(7+3+6+4+4+0)(mod10)\\ \equiv 28+3\left(24\right)(mod10)\\ \equiv 100(mod10)\\ \equiv 0\end{array}$

Since the result is zero, therefore $978-3-16-148410-0$is a valid ISBN - 13 number .

(d) Step 5: Determining that 978−0−201−10179−9 is a valid number.

Given number is

$978-0-201-10179-9$

The check digit $a_{13}$for an ISBN-13 with initial digits $a_1{a}_2\dots a_{12}$ is determined by the congruence

$\left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4\dots +a_{12}\right)\equiv 0(mod10)$

To verify substituting the values of digits

$\begin{array}{r}\equiv (9+8+2+1+0+7+9)+3(7+0+0+1+1+9)(mod10)\\ \equiv 36+3\left(18\right)(mod10)\\ \equiv 90(mod10)\\ \equiv 0\end{array}$

Since the result is zero, therefore $978-0-201-10179-9$is a valid ISBN - 13 number.