A few or few?

We have noticed that a number of our students used few when they meant a few. For example:

(1) ?She has few close friends, and they should be able to help her.

(The question mark ? indicates oddity.)

The word few means very few or none at all. For example:

(2) Even though she has spent more than a year in Tokyo, she knows few words of Japanese.
(3) He is happy because there were few complaints today.

A few, on the other hand, means a small number but not a lot. Yes, that little a makes a difference:

(3) She knows a few words of Japanese, and that proved a great help.
(4) He is unhappy because there were a few complaints today.

Do you see the difference now?

A subtle difference no doubt, but it's what makes sentence (1) semantically odd.

The clause "they should be able to help her" indicates a desirable consequence of having some close friends, and we use a few rather than few to emphasize the bigger number:

(4) She has a few close friends, and they should be able to help her.