An International Standard Book Number (ISBN) is used to identify books. The ISBN number for the 9th edition of The Nature of Mathematics was 0-534-36890-5. The first digit, 0, indicates the book is published in an English-speaking country. The next three digits, 534, identify the publisher (Brooks/Cole), and the next five digits identify the particular book. The last digit is a check digit, which is used as follows. Multiply the first digit by 10, the next digit by 9, and so on:

a. 0-534-13728-x 

b. 0-691-02356-x

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10(0) + 9(5) + 8(3)+7(4) + 6(3) + 5(6) + 4(8) + 3(9) + 2(0)+ 1(5) 209 = This number is congruent to 0, (mod 11). This is not by chance. The check digit, 5, is chosen so that this sum is 0, (mod 11). In other words, the check digit x for the book with ISBN 0-534-34015-x is found by considering 10(0) + 9(5) + 8(3)+7(4) + 6(3) + 5(4) + 4(0) + 3(1) + 2(5) = 148 = 5 This means that for check digit x, 5 + x=0, (mod 11). The one-digit solution to this equation is x = 6, so the check digit is 6. What is the check digit for a book with the given ISBN in Problems 53-54?