Algorithmic Mathematics

Exercise 5.1. Let R Z/(4), and let a = 2r3 + 2.r +3, b = 3r3 + 27% +1+3 be polynomials in R[x]. Determine deg(a?), ldcf(b)-!, ab, a+a, a +b, a MOD b. Exercise 5.2. Let R=Z/(7), and let a = 4.23 + 2x +1, b = 3.r + 1+4 be polynomials in R[u]. Use the algorithm PolynomialQuoRem to determine a DIVb, a MODb, bDIV a, b MOD a. Exercise 5.3. In the statements below, let m and n assume integer values, and let W denote the while-loop. a:=m; b:=n; C:=0; while b + 0 do C:=c+a; b:=b-1; od; (a) For which integer values of n will w terminate? Explain your answer. (6) Prove that ab+c= mn is a loop invariant for W. (c) Suppose that w terminates. Show that on termination we have that c = mn