Chapter 2.7, Problem 12E

Suppose that the check digit is computed as described in Example $5$. Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit.

Example $5$

Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an $11$ the digit may be appended to a $10$-bit identification number to obtain the $11$-digit invoice number of the form

$x_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_7{x}_8{x}_9{x}_{10}c,$ where the $11$ th bit, $c$, is the check digit, computed as

$x_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_7{x}_8{x}_9{x}_{10}\equiv c(\mathrm{mod}n)$.

If congruence modulo $9$ is used, then the check digit for an identification number $3254782201\equiv 7(\mathrm{mod}9)$. Thus the complete correct invoice number would appear as $32547822017$. If the invoice number $31547822017$ were used instead and checked, an error would be detected, since $3254782201\cancel{\equiv }7(\mathrm{mod}9)$.

$[3154782201\equiv 6(\mathrm{mod}9).]$