# ISBN-10/Examples/0-19-853453-1

## 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:

$d = \displaystyle \sum_{k \mathop = 1}^9 k d_k \pmod {11}$

which may or may not need to be translated into $X$ if it equals $10$.

So, we have:

 $\ds \displaystyle \sum_{k \mathop = 1}^9 k d_k \pmod {11}$ $=$ $\ds 1 \times 0 + 2 \times 1 + 3 \times 9 + 4 \times 8 + 5 \times 5 + 6 \times 3 + 7 \times 4 + 8 \times 5 + 9 \times 3 \pmod {11}$ $\ds$ $=$ $\ds 0 + 2 + 27 + 32 + 25 + 18 + 28 + 40 + 27 \pmod {11}$ $\ds$ $=$ $\ds 199 \pmod {11}$ $\ds$ $=$ $\ds 18 \times 11 + 1 \pmod {11}$ $\ds$ $=$ $\ds 1 \pmod {11}$

$\blacksquare$