Clarifying a questionable notion about the proper use of “each”

I’D like to take up a very common word that I myself trip over sometimes because of a notion that got stuck in my mind from way back as pretty well standard and logical. That word is “each,” and that persistent notion is that “each” must always be treated as singular no matter how it is positioned or used in a sentence.

The confusion over its usage stems from the very definition of “each” as an adjective—“being one of two or more distinct individuals having a similar relation and often constituting an aggregate.” This obviously marks any noun preceded by “each” as singular, so expectedly, both the verb and any subsequent pronoun for that noun also need to be singular, as in these sentences: “Each tour guide needs an intensive job orientation.” “Each of the female job applicants was asked about her marital status.” “Each of the commercial establishments has to be equipped with its own CCTV.”