math135-a7

# math135-a7 - p prove that a r â‰ 1(mod p(b Explain why r...

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Assignment 7 – Questions 1. Prove, without using a calculator, that 641 divides the Fermat number F (5) = 2 2 5 + 1 . 2. Find 27 7 9 (mod 11). 3. 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 p ). Write p - 1 = qs + r for some q,r Z with 0 r < s . (a) Starting with a p - 1 1 (mod
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Unformatted text preview: p ), prove that a r â‰¡ 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 â‰¡ 1 (mod 17). 4. In Z 20 , solve the pair of simultaneous equations [7][ x ] + [12][ y ] = [6] [6][ x ] + [11][ y ] = [13] 5. Solve the following system of simultaneous equations in Z 12 . [8][ x ] + [3][ y ] = [9] [6][ x ] + [5][ y ] = [2] 1...
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