Chapter 4, Problem 44SE

Show that there are transpositions of two digit that are not detected by an ISBN-13.

A routing transit number (RTN) is a bank code used in the United States which appears on the bottom of checks. The most common form of an RTN has nine digits, where the last digit is a check digit, if $d_{\text{1}}{d}_{\text{2}}\mathrm{...}{d}_{\text{9}}$ is a valid RTN, the congruence $\text{3}\left(d_{\text{1}}\text{+}{d}_{\text{4}}\text{+}{d}_{\text{7}}\right)\text{ +}\text{7}\left(d_{\text{2}}\text{+ }{d}_{\text{5}}\text{+ }{d}_{\text{8}}\right)+\left(d_{\text{3}}\text{+}{d}_{\text{6}}\text{+}{d}_{\text{9}}\right)\equiv 0\left(\text{mod}\text{1}0\right)$ must hold.