hmwk_8_sol - Discrete Structures Oct 24 2011 Assignment 8:...

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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 ? Solution We note that any number i 0 can be written as i = ap + r , for some positive a and r [0 ,p - 1]. In particular a will be j i p k . Once we have i in this form, it is easy to see i mod p r mod p . (b) If i I , and i < 0 what is i mod p ? Solution As above, we write i = ap + r . If we can write i in such a form for r [0 ,p - 1], then the answer will be the same. The catch, is now that i < 0, a will be negative. Interestingly enough, a will still be j i p k . Try it out! 2. (a) Prove that modular arithmetic under p , ie mod p , is an equivalence relation. Solution For a relation to be an equivalence relation, it must be reflexive, transitive, and sym- metric. Symmetric
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hmwk_8_sol - Discrete Structures Oct 24 2011 Assignment 8:...

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