280hw4 - Computer Science 280: Homework 4 Page 1 of 2...

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Computer Science 280: Homework 4 Homework 4: 2/20/08 (due 2/27/08) The following problems are all taken from the handout on Number Theory from Rosen's book. Section Number Points Comments 2.4 44 3 46 3 2.5 20 4 28 5 Hint: what is the congruence class of 10 mod 11 2.6 2(b),(f) 6 For (f), use the Euclidean algorithm to compute the gcd and then back substitute 8 3 10 4 12 3 15 4 18 3 20 5 Note that you can't immediately apply the CRT, since 6, 10, and 15 are not pairwise relatively prime. You need to convert it to a form where you can apply the CRT. (Hint: if x is congruent to 5 mod 6, what is x congruent to mod 2 and mod 3?) 26 3 28 5 (Hint: for part (b), set up a system of congruences mod 5, 7, and 11 of which 3**302 is a solution; then find another solution. 48 4 There seems to be a bug in Rosen's description of the Euclidean algorithm. Rosen says (just before Exercise 48) that tj = t(j-1) -q(j-1)t(j-2). It should say tj=t(j-1) - qj t(j-2). I'll accept any reasonable backsubstitution approach to computing the coefficients. [A problem from Jon Kleinberg:] [5 points] The folks at MyCrawl are having a bizarre performance problem. As they collect news headlines,they want to know if they've seen a given headline before. They don't want to have to consult all the headlines in their index to do this, so they're using a very primitive form of hashing. Their scheme works as follows. They pick a number n,
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280hw4 - Computer Science 280: Homework 4 Page 1 of 2...

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