MathDB

Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of

10

symbols. Such code looked like this:

a_1a_2 . . . a_9a_{10}

with

a_1, . . . , a_9 \in \{0, 1, . . . , 9\}

and

a_{10} \in \{0, 1, . . . , 9, X\}

. The symbol

X

stood for the number

10

. With a valid ISBN code was

a_1 + 2a2 + . . . + 9a_9 + 10a_{10}

a multiple of

11

. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.

Problem Statement