Problem_Set_07

# Problem_Set_07 - ArsDigita University Month 2: Discrete...

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ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Problem Set 7 – Generating functions, Number Theory, Cryptography 1. Compute by hand, the smallest positive integers x, y, u, v such that ax - by = bu av = gcd( a,b ) for each pair a, b below. Use the Euclidean algorithm and back tracking. Turn in your answers only for a and d. a. 99, 101 b. 10, 35 c. 7, 12 d. 36, 42 2. Write a recursive Scheme function to do the computation above and show the answer for the pair of numbers 233987973 and 41111687. 3. In each of the following expressions, what is the coefficient in front of the term whose exponent is 4? a. ( 1+x+x 2 +x 3 +x 4 ) 3 b. ( 1+x 2 +x 4 ) 2 (1+x+x 2 ) 2 c. ( 1+x+x 2 +x 3 +x 4 +…) 3 4. Find a generating function that will help determine the number of 5-combinations of the letters H, E, L, P in which L and P appear at most once, but H and E can appear multiple times.

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## This document was uploaded on 10/01/2011.

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Problem_Set_07 - ArsDigita University Month 2: Discrete...

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