a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo...

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo n is well defined as a residue class in Zn, and depends only on the residue class a in Zn.

b) Let Z × n ⊆ Zn be the set of invertible residue classes modulo n. Prove that Z × n forms a group under multiplication. Is this group a subgroup of Zn?

c) List the elements of Z × 9 . How many are there? For each residue class u ∈ Z9, compute the elements of the sequence u, u2 , u3 , u4 , . . . until the pattern is clear. Determine the length of each repeating cycle. Is Z × 9 a cyclic group?

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Colby Messinger answered 2 years ago

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