Every book that is published is assigned a unique 10-digit ISBN. For example, the ISBN 0-047024-13-3 is divided into 4 parts: The first, 0, indicates the language group in which the book was published, The second, 047024, identifies the publisher, The third, 13, is a number the publisher assigns to identify a particular book, The fourth, 3, is the check digit k=1 The digit a10 is chosen to satisfy 10 k ak = 0 (mod 11) Question 1 Use a proof by contradiction to show that the ISBN check-digit scheme detects any transposition error. i.e. Assume there were two 10 digit ISBN numbers x and y that are identical except when their mth and nth digits are transposed (for different m, n) and use congruences to reach a contradiction. Question 2 Assume that x = x1234567x8x9x10 is a valid 10 digit ISBN numbers that was printed as a 10 digit number y = y1Y2Y3Y4Y5Y6Y7Y8Y9Y10 that is identical to x, except in the jth position. Show that y is not a valid ISBN number. [Thus, the ISBN check-digit scheme detects any single digit error.]