Translation of "Elle a aucun frère."?
I wrote "She has no brother." but it was not accepted by Duolingo. Possible correct answers given were "She has not got brothers." and "She does not have any brothers."
What? I get over 2 million hits for "she has no brother" (with the quotes) and about 1,1 million for "She doesn't have brothers". It doesn't even make sense to have only 31,000 hits for such a common sentence. I would actually suspect a problem with Google if that were the case.
What's your problem? I'm trying to help you. If you think you know more, don't ask the question. I did the Google experiment again with the same results. I even changed "brother" for sister with no difference.
Explanation: "She has no brother" sounds quite awkward, it only would be natural in a very specific context - correcting someone's mistake. "I saw Mary with her brother yesterday." "She has no brother, you idiot, it was her dad!" In a general context, "She doesn't have any/a brother" is more natural - in English the verb is negated, not the direct object, in most sentences. You wouldn't say "I ate no breakfast", you'd say "I didn't eat (any) breakfast".
I'm sorry if I upset you but you are mistaken. The structure "has no" is used quite commonly. The reason you are not getting many results is because you are using a local version of Google. I just went to Google Chile (where I assume you are) and got similar results as you. Now, if you go to Google.com and click on the bottom link to access Google in English, you'll see the general results without the local bias.
Well, that's my intuition, intuitions are sometimes wrong. Check out google n-grams (which searches millions of books published in English in the last few centuries) and you will see that "does not have" has become more and more popular in the last century while "has no" has declined. It's probable that in spoken language this trend is even stronger, because written language tends to be conservative. In general, learners of English overuse "have no" and underuse "does not have" and it sounds like they walked out of a Jane Austin novel. :)
Intuitions are usually wrong. :)
But could you please paste a link here. The one I see is this: http://books.google.com/ngrams/graph?content=has+no%2Cdoes+not+have&year_start=1800&year_end=2012&corpus=15&smoothing=3&share=
Which shows that even though "does not have" is becoming more popular, is still LESS popular than "has no", which again invalidates your original point of saying that "does not have" is more common than "has no". And without solid evidence, I don't think it is simple to extrapolate from written language to spoken or the other way around. They don't necessarily run together on all aspects of language.