Translation:I have three sisters and no brothers.
I am wondering why the plural form of a noun is needed if you say that you have none of it (geen broers) . In French, for example, it would be considered a mistake if you used the plural in this same example (je n'ai pas de frère, or, je n'ai aucun frère). I am puzzled...
In English you would usually only use the plural 'I don't have any brothers.'/'I have no brothers.' and almost never 'I don't have a brother.'/'I have no brother.' Unless it is something that is assumed to be in the singular, like 'I have no father.'
I am thinking something similar applies here.
I also used that exact answer, like you, I think it is literally correct. But as a native USA English speaker, it did seem awkward- to me it sounds like an intermediate language learner being too wordy. Perhaps that phrasing should remain incorrect simply for that reason, "... and no brothers" is what I would say (if I wished to lie about how many brothers I actually have.... )