assignment-06-algebra

assignment-06-algebra - (b) Determine the value of the...

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MATH 135, Fall 2010 Assignment #6 Due at 4:00pm on Tuesday, November 2 Problem 1 . Most recent books are identified by their International Standard Book Number , or ISBN, which is a 10-digit number, separated into four blocks. The ISBN for our course text book is 0-13-184868-2. Here the first block of digits, 0, represents the language of the book (English), the second block, 13, represents the publisher (Pearson Prentice Hall), the third block, 184868, is the number assigned to the book by the publisher, and the last block, 2, is the check digit . If a 1 ,a 2 , ··· ,a 10 are the 10-digits of the ISBN then for 1 i 9 we have a i ∈ { 0 , 1 , 2 , ··· , 9 } while a 10 ∈ { 0 , 1 , 2 , ··· , 9 ,X } , and X means “10”. The check digit a 10 is used to determine whether an error has been made when an ISBN is copied. It is chosen so that 10 X i =1 ia i 0 (mod 11) . (a) Determine whether the number 5-22-390112-X is a valid ISBN.
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Unformatted text preview: (b) Determine the value of the digit a such that the number 2-17-8a2250-3 is a valid ISBN. (c) When an ISBN was copied, two adjacent digits were interchanged resulting in the number 1-18-397604-5. Determine the original ISBN. Problem 2 . A positive integer is divisible by 8 if and only if the three-digit integer formed by its nal three digits is divisible by 8. (You do not need to prove this.) Determine all pairs of digits ( a,b ) such that 72 b 9 a 2 is divisible by 88. Problem 3 . Use mathematical induction to show that 3 n 1+2 n 2 (mod 8) for all integers n 0. Problem 4 . Suppose that a,b Z and m,n P . Prove that an bn (mod mn ) if and only if a b (mod m ). 1...
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