Math 1600 Section 1.4 Question 35
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The solutions manual gets (b) and (c) wrong, so I'll give
a sketch of the solution here. Please try to do the problem
on your own before reading the solution (not just for this
problem, but in general).
Hint:
The check vector for ISBN-10 is $\vc = [10,9,8,7,6,5,4,3,2,1]$.
When you transpose two consecutive digits $a$, $b$ in an ISBN-10 code,
the dot product changes by $b - a$. (Convince yourself of this!)
This makes (b) and (c) easy to solve.
Solution:
Recall that $\vu = [0,8,3,7,0,9,9,0,2,6]$.
(a) The dot product $\vc \cdot \vu = 5 \pmod{11}$, so $\vu$ is not a
valid ISBN-10 code.
(b) We want to transpose two digits of $\vu$ so that the dot product
goes down by 5 or up by 6. So, using the hint, we want consecutive
digits $a$ and $b$ such that $b - a = -5$ or $6$. The only digits
that make this happen are the second and third, since $3 - 8 = -5$.
(c) Choose any invalid ISBN-10 code where there are two pairs of
consecutive digits with the correct difference. For example,
start with $[4,3,2, \ldots]$ and complete it to a code word
where the dot product is 1 modulo 11. Then exchanging either
the 4 and the 3 or the 3 and the 2 will produce a valid code.
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