APL (Dyalog), 50 bytes
Look 'ma, no built-in totient!
0{⍵=1:⍺⋄(⍺+1)∇×/⍵,1-÷o/⍨0=⍵|⍨o←(⊢~∘.×⍨)1+⍳⍵}¨1+⍳99 How?
(⊢~∘.×⍨)1+⍳⍵ - primes (better don't ask)
o/⍨ - filter by
0=⍵|⍨o - divisibility by n
1-÷o - array wise 1- recipeocal
×/⍵, - prepend n, and multiply 'em all
This is the totient. All the rest is wrapper for the counting and applying on 2..100.