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# hmwk_8 - 4 3 3 mod 4 one may do 2 4 3 3 mod 4 =(8(9 mod 4 =...

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Discrete Structures Oct 24 2011 Assignment 8: Due at beginning of class Monday, Oct 24th Prof. Hopcroft *******Note: this homework is subject to point evaluations for clarity of writing and clarity of mathematical statements.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. Consider the integers, I = { ..., - 2 , - 1 , 0 , 1 , 2 ,... } , and a prime p. (a) if, i I , and i 0 what is i (mod p )? (b) if i I , and i < 0 what is i (mod p )? 2. (a) Prove that modular arithmetic under p , ie mod p , is an equivalence relation. (b) Note that in modular arithmetic one represents each equivalence class by a representative of the class and deﬁnes arithmetic for the representative elements. Eg, in mod 4, the representative elements are 0 , 1 , 2 , 3. If p = 3 what are the addition and multiplication tables for the representative elements? 3. In modular arithmetic some people get sloppy and use a sequence of operations that produce numbers greater than the mod, we are working in. For example in computing 2
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Unformatted text preview: * 4 + 3 * 3 mod 4, one may do: 2 * 4 + 3 * 3 mod 4 = (8) + (9) mod 4 = 17 mod 4 = 1 mod 4 , rather than: 2 * 4 + 3 * 3 mod 4 = (0) + (1) mod 4 = 1 mod 4 . Is it okay to produce numbers larger than the mod we are working in and if so why? 4. How many ways can one write seven as the sum of four nonnegative integers? (Note: Order does not matter, 6+1+0+0 is the same as 0+0+1+6, don’t count it twice.) 5. (a) Roll a k-sided dice, three times: How many possible outcomes are there? What is the probability of a face appearing exactly two times? (b) Roll a 6-sided dice three times: What is the probability of each face being diﬀerent in three rolls? What is the probability of a face appearing exactly two times? What is the probability of a face appearing three times? What is the sum of the above three probabilities? 1...
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## This note was uploaded on 11/11/2011 for the course CS 2800 at Cornell.

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