Let D18 denote the set of positive divisors of 18. For d ∈ D18 let Sd = {n|0 < n ≤ 18 and gcd(n, 18) = d}.
(a) Show that the collection Sd, d ∈ D18, provides a partition of {1, 2, 3, 4, ... , 17, 18}.
(b) Note that |S1| = 6 = ϕ(18) and |S2| = 6 = ϕ(9). For each d ∈ D18, express |Sd| in terms of Euler's phi function.