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All books are identified by an International Standard Book Number (ISBN), a 10–digit code $x_1,x_2,\cdots,x_{10}$, assigned by the publisher. (This system was changed in 2007 when a new 13–digit code was introduced.) These 10 digits consists of blocks identifying the language, the publisher, the number assigned to the book by its publishing company, and finally, a 1–digit check digit that is either a digit or the letter $X$ (used to represent 10).The check digit is selected so that the sum of $\sum_{i=1}^{10} (i\cdot x_i) \equiv 0 \pmod{11}$ and is used to detect errors in individual digits and transposition of digits.

The ISBN of the fifth edition of Elementary Number Theory and Its Applications is 0-32-123Q072, where $Q$ is a digit. How do I find out the value of $Q$?

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kyla
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    Compute sum of other digits (multiplied by rank), modulo 11. What do you have to add to get 0 mod 11? – Jean-Claude Arbaut Mar 25 '13 at 14:30
  • I'm confuse, the question is too long. Can you make it clear in a easy way to compute Q. – kyla Mar 25 '13 at 14:36
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    Do as if you were computing check digit, only $4Q$ will remain unknown, and you know the sum must be 0 mod 11. You will have an equation $4Q+K=0 \mod 11$, with $0 \leq Q \leq 9$. See wikipedia for an example. It may also help to know that $4 \cdot 3 = 1 \mod 11$, to finally compute $Q$. – Jean-Claude Arbaut Mar 25 '13 at 14:38

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