Practice question · Numerical answer
An ISBN-10 has digits d1 through d10 and must satisfy: 1 x d1 + 2 x d2 + ... + 10 x d10 is congruent to 0 mod 11. The first nine digits are 0, 3, 0, 6, 4, 0, 6, 1, 5. Find the check digit d10.
Hints
- Compute the weighted sum of the first nine digits, then reduce it modulo 11.
- The weighted sum is 145, and 145 = 11 x 13 + 2. Note that the weight 10 is congruent to -1 mod 11.
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2
Why
The weighted sum of the first nine digits is 6 + 24 + 20 + 42 + 8 + 45 = 145, which is 2 mod 11. Since the tenth weight 10 is congruent to -1, we need 2 - d10 congruent to 0, so d10 = 2. (This is the real ISBN 0-306-40615-2.) Changing any single digit breaks the congruence, which is exactly how the check catches errors.
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