Check digits

A few weeks ago I was looking at how ISBN numbers are constructed and read that the last digit in the ISBN number is a so called check digit used to check for errors in the preceding digits. Check digits are also used in bank account numbers for example. The check digits are generated by an algorithm. For ISBN there is only 1 check digit and the algorithm used is different for ISBN 10 and ISBN 13 (in use since January 2007).

For ISBN 10 the check digit computed by multiplying each digit with its position in the number (counting from the right) and taking the sum of these products modulo 11 is 0. The digit the farthest to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct. It may need to have the value 10, which is represented as the letter X. Applied to ISBN 0-201-53082-1 this gives : 0×10 + 2×9 + 0×8 + 1×7 + 5×6 + 3×5 + 0×4 + 8×3 + 2×2 + 1×1 = 99 ≡ 0 (mod 11). So the ISBN is valid. The positions can also be counted from left, in which case the check digit is multiplied by 10, to check validity: 0×1 + 2×2 + 0×3 + 1×4 + 5×5 + 3×6 + 0×7 + 8×8 + 2×9 + 1×10 = 143 ≡ 0 (mod 11).

For ISBN 13 the number is the same as the EAN 13 number where EAN stands for European Article Number. It uses an algorithm that is similar to the one used for the so called universal product code. The calculation is as follows : add the digits in the odd-numbered positions from the right (first, third, fifth, etc. – not including the check digit) together and multiply by 3. Add the digits in the even-numbered positions together and add to the previously obtained result. Determine the modulo 10 of this number. If it is 0 then the check digit is 0, if it is not, subtract it from 10 to get the check digit. Applied to ISBN 978-90-8803-092-5 this gives : (2 + 0 + 0 + 8 + 9 + 7)*3 + (9 + 3 + 8 + 0 + 8 + 9) = 26*3 + 37 = 78 + 37 = 115. 115 modulo 10 =5. Check digit is 10 – 5 = 5.