Mark/handouts
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Mark 2024-04-01 07:21:08 -07:00
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Advanced/Error-Correcting Codes/parts/00 detection.tex
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@ -8,7 +8,7 @@ Say we have a sequence of nine digits, forming a partial ISBN-10: $n_1 n_2 ... n
The final digit, $n_{10}$, is chosen from $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ so that: The final digit, $n_{10}$, is chosen from $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ so that:
$$ $$
\sum_{i = 1}^{10} (11 - i)n_i \sum_{i = 1}^{10} (11 - i)n_i \equiv 0 \text{ mod } 11
$$ $$
If $n_{10}$ is equal to 10, it is written as \texttt{X}. If $n_{10}$ is equal to 10, it is written as \texttt{X}.
@ -31,7 +31,7 @@ Only one of the following ISBNs is valid. Which one is it?
\problem{} \problem{}
Take a valid ISBN-10 and change one digit. Is it possible that you get another valid ISBN-10? \par Take a valid ISBN-10 and change one digit. Is it possible that you get another valid ISBN-10? \par
Provide an example or a proof. Provide a proof.
\begin{solution} \begin{solution}
Let $S$ be the sum $10n_1 + 9n_2 + ... + 2n_9 + n_{10}$, before any digits are changed. Let $S$ be the sum $10n_1 + 9n_2 + ... + 2n_9 + n_{10}$, before any digits are changed.