Elementary Number Theory, ISBN numbers and mathematics olympiads

Question 1:

The International Standard Book Number (ISBN) used in many libraries consists of nine digits a_{1} a_{2}\ldots a_{9} followed by a tenth check digit a_{10} (somewhat like Hamming codes), which satisfies

a_{10} = \sum_{k=1}^{9}k a_{k} \pmod {11}

Determine whether each of the ISBN’s below is correct.
(a) 0-07-232569-0 (USA)
(b) 91-7643-497-5 (Sweden)
(c) 1-56947-303-10 (UK)

Question 2:

When printing the ISBN a_{1}a_{2}\ldots a_{9}, two unequal digits were transposed. Show that the check digits detected this error.

Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.

More later,
Nalin Pithwa.

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