"Nie mam pięciu krzeseł."
Translation:I do not have five chairs.
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Usually when we talk about a number 'going with' some case, we're considering it 'on its own', as a standalone phrase, outside of a sentence. Things can change if we put it in a sentence.
"pięć", like most numerals, takes Genitive even as a standalone phrase. However, I guess in this sentence it would be better to say that it's in Genitive because of "nie mam".
"trzy" does not take Genitive, let's say it's neutral (term suggested by one of the learners here): it just takes whatever case is needed by the context. So as a standalone phrase it's "trzy krzesła", in Nominative. But here it would be "Nie mam trzech krzeseł" anyway, due to "nie mam". But if the sentence was "I have three chairs", then it's "Mam trzy krzesła", so it's different than "Mam pięć krzeseł" which already has Genitive 'inside' it.