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COT3100Hmk05 - Instead use the binomial theorem and any mod...

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COT 3100 Fall 2002 Homework #5 Assigned: 10/24/02 Due: 11/5/02 in lecture 1) a) Use Euclid's Algorithm to find the greatest common divisor of 962 and 629. b) Use the Extended Euclidean Algorithm to find integers x and y such that 962x+629y = gcd(962, 629). 2) Let x and y be integers. If 13 | (2x+5y), prove there are no integer solutions to 3x+y = 2003. 3) Given the prime factorization of n is p a q b , where p and q are prime and a and b are positive integers, determine the number of factors of n. (Hint: Note that each factor of n MUST only contain p and q in its prime factorization, and can not contain more than a p's or b q's.) As an example, find the total number of factors of 108 and list each of these. This is a counting question...) 4) Prove that for all integers a, 16 | ((2a+1) 8 - 1). Do NOT use induction for this proof.
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Unformatted text preview: Instead, use the binomial theorem and any mod rules you deem necessary. 5) If A ={ 1, 2, 3, 4, 5, 6, 7, 8 }, determine the number of relations on A that are (a) reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain ( 1 , 2 ); (e) symmetric and contain ( 1 , 2 ); (f) anti-symmetric; (g) anti-symmetric and contain ( 1 , 2 ); (h) symmetric and anti-symmetric; (i) reflexive, symmetric and anti-symmetric. 6) Prove or disprove: Let R be a relation over A x A. If R ° R is transitive, then R is transitive as well. 7) Consider the following relation R defined over the set of positive integers: R = {(x,y) | x/y = 4 ∨ y/x = 4} Determine if the relation R is (i)reflexive, (ii)irreflexive, (iii)symmetric, (iv)anti-symmetric, and (v)transitive....
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