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Math 4330, Homework 5, 2/19/2014 : Due 2/24/2014 1 1. Show that if u > v then (u mod v) ≤ u/2. 2. Use the previous result to prove that in Algorithm A, Step A2 is executed no more than 1 + 2 log2 u times. (In particular, it follows that the Euclidean Algorithm is polynomial-time in the input size which is log2 u + log2 v.) 3. Implement Algorithm X verbatim (i.e., name your variables the same way and use comments to label the steps X1, X2, and X3). Use it to find an integer x such that 171x ≡ 1 (mod 10000000000001) . 1 c This document is copyright 2014 Chris Monico, and may not be reproduced in any form without written permission from the author. 1