how to determine the gcd of following

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I am stuck at a task... I want to determine $n$ of those terms:

$ 1) gcd(n^{1000}(n^2-1))$

$ 2) gcd(n^{1000}(n^4-1)) $

$ 3) gcd(n^{1000}(n^6-1))$

$4) gcd(n^{1000}(n^{12} -1)) $

where $ n \in \mathbb{N} $

What I think the task is about,is to determine the gcd of all numbers of the form f.e $n^{1000}(n^2-1) $ and so on.. How can I do that?

appreciate any help!