Question

Show that the check digit of an can always detect

a single error.

Short answer

It is shown that the check digit of an can always detect a single error.

Step-by-step solution

Mutually relatively prime

A set of integers is called mutually relatively prime if the greatest common divisor of these integers is 1.

Identifying the data given in the question

Given in the question,

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

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

Proving that the check digit of an ISBN-13 can always detecta single error.

It is given$a_1,a_2.a_{12}$ are digits.

Suppose a single error occurred and$a_i$ is changed to$x_i$ .

when i is even

Then,

$\begin{array}{r}\equiv \left(a_1+a_3\dots +a_{13}\right)+3\left(a_2+a_4+\dots +\left(a_i-x_i\right)\dots +a_{12}\right)(mod10)\\ \equiv 3\left(a_i-x_i\right)(mod10)\end{array}$

Since$a_j\ne x_j$ and 3 and 10 are mutually relatively prime.

So, $3\left(a_i-x_i\right)(mod10)\ne 0$

Therefore, a single error will be detected.

Now, suppose is odd

Then

$\begin{array}{r}\equiv \left(a_1+a_3+\dots \left(a_i-x_i\right)+\dots +a_{13}\right)+3\left(a_2+a_4+\dots +a_{12}\right)(mod10)\\ \equiv \left(a_i-x_i\right)(mod10)\end{array}$

Since$a_j\ne x_j$ , and 1 and 10 are mutually relatively prime.

So,$\left(a_j-x_j\right)(mod10)\ne 0$

Therefore, a single error will be detected.

Hence, we can conclude that the check digit of a can always detect

a single error.