ISBN-10/Examples/0-19-853453-1
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Example of ISBN-$10$
The book whose ISBN-$10$ code begins:
- $0$-$19$-$853453$
has a check digit of $1$.
Proof
The check digit $d$ of an ISBN-$10$ is calculated by definition as:
- $d = \ds \sum_{k \mathop = 1}^9 k d_k \pmod {11}$
which may or may not need to be translated into $X$ if it equals $10$.
So, we have:
\(\ds \sum_{k \mathop = 1}^9 k d_k \pmod {11}\) | \(=\) | \(\ds 1 \times 0 + 2 \times 1 + 3 \times 9 + 4 \times 8 + 5 \times 5 + 6 \times 3 + 7 \times 4 + 8 \times 5 + 9 \times 3 \pmod {11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 + 2 + 27 + 32 + 25 + 18 + 28 + 40 + 27 \pmod {11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 199 \pmod {11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 18 \times 11 + 1 \pmod {11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \pmod {11}\) |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes