Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14COMPSCI 102Introduction to Discrete MathematicsRules of the GameEach person will have a unique numberFor each question, I will first give the class time to work out an answer. Then, I will call three different people at randomThey must explain the answer to me. If I’m satisfied, the class gets points. If the class gets 1,700 points, then you winGCDDEFINITIONALGOPROPERTIESGCDALGORITHMCONVER-GENTSEXAMPLESRUNNING TIMEANALGORITHM400300200100100200300400CONTINUEDFRACTIONS1. The Greatest Common Divisor (GCD) of two non-negative integers A and B is defined to be:2. As an example, what is GCD(12,18) and GCD(5,7)The largest positive integer that divides both A and BGCD(12,18) = 6GCD(5,7)=1A Naïve method for computing GCD(A,B) is:Give an algorithm to compute GCD(A,B) that does not require factoring A and B into primes, and does not simply try dividing by most numbers smaller than A and B to find the GCD. Run your algorithm to calculate GCD(67,29)Factor A into prime powers. Factor B into prime powers.Create GCD by multiplying together each common prime raised to the highest power that goes into both A and B.Euclid(A,B) = Euclid(B, A mod B)Stop when B=0Euclid’s GCD algorithm can be expressed via the following pseudo-code: Euclid(A,B)If B=0 then return Aelse return Euclid(B, A mod B)Show that if this algorithm ever stops, then it outputs the GCD of A and BGCD(A,B) = GCD(B, A mod B)( d | A and d | B ) ( d | (A - kB ) and d | B )The set of common divisors of A, B equalsthe set of common divisors of B, A - kBProof:Show that the running time for this algorithm is bounded above by 2log2(max(A,B))Proof:Claim: A mod B < ½ AIf B = ½ A then A mod B = 0If B < ½ A then any X Mod B < B < ½ AIf B > ½ A then A mod B = A - B < ½ AEuclid(A,B) = Euclid(B, A mod B)Stop when B=0GCD(A,B) calls GCD(B, <½A)Proof of Running Time:which calls GCD(<½A, B mod <½A)Every two recursive calls, the input numbers drop by halfDAILY DOUBLEA simple continued fraction is an expression of the form:a +b +c +d +e + …1111Where a,b,c,d,e, … are non-negative integers. We denote this continued fraction by [a,b,c,d,e,…]. What number do the fractions [3,2,1] and [1,1,1] represent? (simplify your answer)Let r1 = [1] r2 = [1,1] r3 = [1,1,1] r4 = [1,1,1,1]::Find the value of rn as a ratio of somethig we’ve seen before (prove your answer)rn = Fib(n+1)/F(n) Fib(n+1)Fib(n)Fib(n)+Fib(n-1)Fib(n)= 1 + 1 Fib(n)Fib(n-1)=Let = [a1, a2, a3, ...] be a continued fractionDefine: C1 = [a1] C2 = [a1,a2] C3 = [a1,a2,a3] :Ck is called the k-th convergent of is the limit of the sequence C1, C2, C3,…A rational p/q is the best approximator to a real if no rational number of denominator smaller than q comes closer to Given any CF representation of , each con-vergent of the CF is a best approximator for 1317115111292111111121 ....p = ++++++++++C1 = 3C2 = 22/7C3 = 333/106C4 = 355/113C5 = 103993/33102C6 =104348/33215 Find best approximators for with denominators 1, 7 and 1131. Write a continued fraction for 67/29 2 +3 +4 +21112. Write a formula that allows you to calculate the continued fraction of A/B in 2log2(max(A,B)) steps1modA ABB BA B� �= +� ��
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