sol1 - CMPSC 465 SOLUTIONS TO ASSIGNMENT 1 This assignment...

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CMPSC 465 SOLUTIONS TO ASSIGNMENT 1 This assignment is worth a total of 100 points Fall 2008 1. [14 Points] (a) We are to compute gcd(46139 , 47027) by applying Euclid’s algorithm, that is via a series of integer divisions with remainders gcd(46139 , 47027) 46139 = 47027 × 0 + 46139 = gcd(47027 , 46139) 47027 = 46139 × 1 + 888 = gcd(46139 , 888) 46139 = 888 × 51 + 851 = gcd(888 , 851) 888 = 851 × 1 + 37 = gcd(851 , 37) 851 = 37 × 23 + 0 = gcd(37 , 0) = 37 [Canceled] (b) Compare with the algorithm that checks consecutive integers from 46139 to 37. Each of the 46139 - 36 = 46133 integers in this range will be tested as divisor of both 46139 and 47027, which requires one or two divisions. The total number of divisions thus lies between 46133 and 2 × 46133 = 92266 . 2. [12 Points] We are to write the version of Euclid’s algorithm that uses only subtractions instead of divisions. There are several ways of doing this. For example: Algorithm Euclid 2( m, n ) // Input: Integers m, n 0 // Output: gcd( m, n ) while n = 0 do if m < n then swap ( m, n ) ; // temp m ; m n ; n temp m m - n ; return m Alternatively, if both input numbers are assumed to be positive, then a 0 can only occur as di ff erence of two equal terms, and we could stop there: Algorithm Euclid 3( m, n ) // Input: Integers m, n > 0 // Output: gcd( m, n ) while m = n do if m > n then m m - n else n n - m ; return m (This algorithm gets stuck if one of the numbers is 0, which is why it requires input m, n > 0 or an additional first statement handling the case " a = 0 or b = 0 ".) 3. [Not graded] Begin by making
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