a) Suppose that a ? Z is a unit modulo n. Prove that its inversemodulo n is well defined as a residue class in Zn, and depends onlyon the residue class a in Zn.
b) Let Z × n ? Zn be the set of invertible residue classesmodulo n. Prove that Z × n forms a group under multiplication. Isthis group a subgroup of Zn?
c) List the elements of Z × 9 . How many are there? For eachresidue class u ? Z9, compute the elements of the sequence u, u2 ,u3 , u4 , . . . until the pattern is clear. Determine the length ofeach repeating cycle. Is Z × 9 a cyclic group?
