M135F08A7

# M135F08A7 - MATH 135 Assignment #7 Hand-In Problems 1....

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MATH 135 Fall 2008 Assignment #7 Due: Wednesday 05 November 2008, 8:20 a.m. Hand-In Problems 1. Suppose that a = ( r n r n - 1 ··· r 2 r 1 r 0 ) 10 . This notation means that a = 10 n r n + 10 n - 1 r n - 1 + ··· + 10 r 1 + r 0 where each of r n , r n - 1 , . . . , r 1 , r 0 is a digit between 0 and 9, inclusive. Prove that 11 | a if and only if 11 | r 0 - r 1 + r 2 - r 3 + ··· + ( - 1) n r n . 2. A positive integer is divisible by 8 if and only if the three-digit integer formed by its ﬁnal three digits is divisible by 8. (You do not need to prove this.) Determine all pairs of digits ( a, b ) such that 23 b 2 a 4 is divisible by 72. 3. In each part, determine if the congruence has solutions. If it does, determine the complete solution. (a) 1713 x 851 (mod 2000) (b) 1426 x 851 (mod 2000) (c) x 2 2 x (mod 12) (d) 8 x 12 (mod 52) 4. (a) Determine the inverse of [23] in Z 71 . (b) Determine the integer a with 0 a < 71 such that [ a ] = [3] + [23] - 1 [10] in Z 71 . 5. (a) Prove that 3

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## This note was uploaded on 09/04/2009 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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M135F08A7 - MATH 135 Assignment #7 Hand-In Problems 1....

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