"We take the mice off them."
Translation:Bainimid na lucha díobh.
I find this to be confusing due to the sentence itself. In Ireland to take 'something' off someone means to take it away from them. Given that's the case , I fail to see why Tógaimid + uathu can't be used. I take the book from you is = Tógaim an leabhar uait. Bainim from my understanding would be to remove.
An and Na are the singular and plural definite article. English doesn't differentiate between the singular and plural definite article, but other languages do ("le" vs "les" in French, for example).
The noun itself also has a singular and a plural form - an buachaill / na buachaillí but that's a separate issue from the article (because there is no indefinite article, you can just rely on the article to indicate whether the noun is singular or plural).
The patterns used to form the plural forms of nouns are indeed confusing, ach sin scéal eile.
(There is one slight wrinkle with an and na - when a feminine noun is used in the genitive, na is used for the singular definite article - i lár na páirce - "in the middle of the field").
I don't think there are really any rules regarding plurals, they seem quite irregular across the board. I wish the course creators would do better at presenting the singular forms using "an" and plural forms using "na" at the same time right off the bat (like in the hints drop down when they introduce a new word), but for the most part the only thing I've noticed is that, overall, changing a singular noun using "an" to a plural noun using "na" means the addition of at least one letter in the spelling of the noun. Examples: an leabhar = na leabhair, an anmhi = na anmhithe, etc. I'm very bad at retaining technical rules of grammar, but in terms of patterns that's what I've observed.