MAT135-Assignment7

MAT135-Assignment7 - p Write p-1 = qs r for some q,r ∈ Z...

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MATH 135, Fall 2010 Assignment #7 Due at 4:00pm on Tuesday, November 9 Problem 1 . Suppose that a = ( r n r n - 1 ··· r 2 r 1 r 0 ) 10 3 . This notation means that a = 10 3 n r n + 10 3( n - 1) r n - 1 + ··· + 10 3 r 1 + r 0 where each of r n ,r n - 1 ,...,r 1 ,r 0 is a 3-digit number between 0 and 999, inclusive. Prove that 13 | a if and only if 13 | r 0 - r 1 + r 2 - r 3 + ··· + ( - 1) n r n . Problem 2 . Let p be a prime number and a a positive integer not divisible by p . Fermat’s Little Theorem tells us a p - 1 1 (mod p ). But p - 1 might not be the smallest positive integer k for which a k 1 (mod p ). Suppose that s is the smallest positive integer for which a s 1 (mod
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Unformatted text preview: p ). Write p-1 = qs + r for some q,r ∈ Z with 0 ≤ r < s . (a) Starting with a p-1 ≡ 1 (mod p ), prove that a r ≡ 1 (mod p ). (b) Explain why r must equal 0. (c) Explain why s | ( p-1). (d) Find the smallest positive integer s for which 8 s ≡ 1 (mod 23). Problem 3 . (a) Find 41 515 (mod 17). (c) Find 100 ∑ k =1 k k ! (mod 11). Problem 4 . In Z 20 , solve the pair of simultaneous equations [3][ x ] + [5][ y ] = [6] [5][ x ] + [7][ y ] = [14] 1...
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This note was uploaded on 12/10/2010 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

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