Let n be an integer such that n = p i q j for two distinct prime numbers p and q, and integers i ≥ 1 and j ≥ 1. (For example, we can write 544 = 171 · 2 5 ; here p = 17, q = 2, i = 1, and j = 5.) Let...

Let n be an integer such that n = p i q j for two distinct prime numbers p and q, and integers i ≥ 1 and j ≥ 1. (For example, we can write 544 = 171 · 2 5 ; here p = 17, q = 2, i = 1, and j = 5.) Let P := {k ∈ {1, . . . , n} : p | k} and Q := {k ∈ {1, . . . , n} : q | k}. Argue that ϕ(n) = n(1 − 1 p )(1 − 1 q ) by using Inclusion–Exclusion to compute |P ∪ Q|. (You should find the last two exercises helpful.)